Local lagged adapted generalized method of moments dynamic process

ABSTRACT

Aspects of a local lagged adapted generalized method of moments (LLGMM) dynamic process are described herein. In one embodiment, the LLGMM process includes obtaining a discrete time data set as past state information of a continuous time dynamic process over a time interval, developing a stochastic model of the continuous time dynamic process, generating a discrete time interconnected dynamic model of local sample mean and variance statistic processes (DTIDMLSMVSP) based on the stochastic model, and calculating a plurality of admissible parameter estimates for the stochastic model using the DTIDMLSMVSP. Further, in some embodiments, the process further includes, for at least one of the plurality of admissible parameter estimates, calculating a state value of the stochastic model to gather a plurality of state values, and determining an optimal admissible parameter estimate among the plurality of admissible parameter estimates that results in a minimum error among the plurality of state values.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.62/068,848, filed Oct. 27, 2014, the entire contents of which are herebyincorporated herein by reference. The application also claims thebenefit of U.S. Provisional Application No. 62/246,189, filed Oct. 26,2015, the entire contents of which are hereby incorporated herein byreference.

GOVERNMENT LICENSE RIGHTS

This invention was made with government support under Grant NumbersW911NF-12-1-0090 and W911NF-15-1-0182 awarded by the Army ResearchOffice. The government has certain rights in the invention.

BACKGROUND

Tools for analyzing and managing large collections of data are becomingincreasingly important. For example, data models between variouscommodities can be analyzed to determine whether a collaborative orcompetitive relationship exists between the commodities. However,traditional methods of verifying and validating nonlinear time seriestype data sets can encounter state and parameter estimation errors.

BRIEF DESCRIPTION OF THE DRAWINGS

Many aspects of the present disclosure can be better understood withreference to the following drawings. The components in the drawings arenot necessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the present disclosure. Further, in thedrawings, like reference numerals designate corresponding partsthroughout the several views.

FIG. 1 illustrates an example computing environment for a local laggedadapted generalized method of moments dynamic process according tovarious aspects of the embodiments described herein.

FIG. 2 illustrates a local lagged adapted generalized method of momentsdynamic process according to various aspects of the embodimentsdescribed herein.

FIG. 3 illustrates a process of generating a discrete timeinterconnected dynamic model of statistic processes in the process shownin FIG. 2 according to various aspects of the embodiments describedherein.

FIG. 4A illustrates real and simulated prices for natural gas using thelocal lagged adapted generalized method of moments dynamic processaccording to various aspects of the embodiments described herein.

FIG. 4B illustrates real and simulated prices for ethanol using thelocal lagged adapted generalized method of moments dynamic processaccording to various aspects of the embodiments described herein.

FIG. 5A illustrates real and simulated U.S. treasury bill interest ratesusing the local lagged adapted generalized method of moments dynamicprocess according to various aspects of the embodiments describedherein.

FIG. 5B illustrates real and simulated U.S. eurocurrency exchange ratesusing the local lagged adapted generalized method of moments dynamicprocess according to various aspects of the embodiments describedherein.

FIG. 6A illustrates the real, simulated, forecast, and 95% limit naturalgas spot prices using the local lagged adapted generalized method ofmoments dynamic process according to various aspects of the embodimentsdescribed herein.

FIG. 6B illustrates the real, simulated, forecast, and 95% limit ethanolprices using the local lagged adapted generalized method of momentsdynamic process according to various aspects of the embodimentsdescribed herein.

FIG. 7A illustrates the real, simulated, forecast, and 95% limit U.S.treasury bill interest rates using the local lagged adapted generalizedmethod of moments dynamic process according to various aspects of theembodiments described herein.

FIG. 7B illustrates the real, simulated, forecast, and 95% limit U.S.Eurodollar exchange rates using the local lagged adapted generalizedmethod of moments dynamic process according to various aspects of theembodiments described herein.

FIG. 8 illustrates an example schematic block diagram of the computingdevice 100 shown in FIG. 1 according to various embodiments describedherein.

The drawings illustrate only example embodiments and are therefore notto be considered limiting of the scope described herein, as otherequally effective embodiments are within the scope and spirit of thisdisclosure. The elements and features shown in the drawings are notnecessarily drawn to scale, emphasis instead being placed upon clearlyillustrating the principles of the embodiments. Additionally, certaindimensions may be exaggerated to help visually convey certainprinciples. In the drawings, similar reference numerals between figuresdesignate like or corresponding, but not necessarily the same, elements.

DETAILED DESCRIPTION

1. Introduction

The embodiments described herein are directed to the development andapplication of a local lagged adapted generalized method of moments(LLGMM) dynamic process. Various embodiments of the approach can includeone or more of the following components: (1) developing a stochasticmodel of a continuous-time dynamic process, (2) developing one or morediscrete time interconnected dynamic models of statistic processes, (3)utilizing Euler-type discretized schemes for non-linear andnon-stationary systems of stochastic differential equations, (4)employing one or more lagged adaptive expectation processes fordeveloping generalized method of moment/observation equations, (5)introducing conceptual and computational parameter estimation problems,(6) formulating a conceptual and computational state estimation scheme,and (7) defining a conditional mean square ϵ-best sub-optimal procedure.

The development of the LLGMM dynamic process is motivated by andapplicable to parameter and state estimation problems in continuous-timenonlinear and non-stationary stochastic dynamic models in biological,chemical, engineering, energy commodity markets, financial, medical,physical and social science, and other fields. The approach result in abalance between model specification and model prescription ofcontinuous-time dynamic processes and the development of discrete timeinterconnected dynamic models of local sample mean and variancestatistic processes (DTIDMLSMVSP). DTIDMLSMVSP is the generalization ofstatistic (sample mean and variance) for random sample drawn from thestatic dynamic population problems. Further, it is also an alternativeapproach to the generalized autoregressive conditionalheteroskedasticity (GARCH) model, and it provides an iterative schemefor updating statistic coefficients in a system of generalized method ofmoment/observation equations. Furthermore, the application of the LLGMMto various time-series data sets demonstrates its performance inforecasting and confidence-interval problems in applied statistics.

Most existing parameter and state estimation techniques are centeredaround the usage of either overall data sets, batched data sets, orlocal data sets drawn on an interval of finite length T. This leads toan overall parameter estimate on the interval of length T. Theembodiments described herein apply a new approach, the LLGMM. The LLGMMis based on a foundation of: (a) the Itô-Doob Stochastic Calculus, (b)the formation of continuous-time differential equations for suitablefunctions of dynamic state with respect to original SDE (using Itô-Doobdifferential formula), (c) constructing corresponding Euler-typediscretization schemes, (d) developing general discrete timeinterconnected dynamic model of local sample mean and variance statisticprocesses (DTIDMLSMVSP), (e) the fundamental properties of solutionprocess of system of stochastic differential equations, for example:existence, uniqueness, continuous dependence of parameters.

One of the goals of the parameter and state estimation problems is formodel validation rather than model misspecification. For continuous-timedynamic model validation, existing real world data sets are utilized.This real world data is time varying and sampled, drawn, or recorded atdiscrete times on a time interval of finite length. In view of this,instead of using an existing econometric specification/Euler-typenumerical scheme, a stochastic numerical approximation scheme isconstructed using continuous time stochastic differential equations forthe LLGMM process described herein.

In almost all real world dynamic modeling problems, future states ofcontinuous time dynamic processes are influenced by past state historyin connection with response/reaction time delay processes influencingthe present states. That is, many discrete time dynamic models depend onthe past state of a system. The influence of state history, the conceptof lagged adaptive expectation process, and the idea of a moving averagelead to the development of the general DTIDMLSMVSP. Extensions of thediscrete time sample mean and variance statistic processes are: (a) toinitiate the use of a discrete time interconnected dynamic approach inparallel with the continuous-time dynamic process, (b) to shorten thecomputation time, and (c) to significantly reduce state error estimates.

Utilizing the Euler-type stochastic discretization, for example, of thecontinuous time stochastic differential equations/moment/observationsand the discrete time interconnected dynamic approach in parallel withthe continuous-time dynamic process (and the given real world timeseries data and the method of moments), systems of localmoment/observation equations can be constructed. Using the DTIDMLSMVSPand the lagged adaptive expectation process for developing generalizedmethod of moment equations, the notions of data coordination,theoretical iterative and simulation schedule processes, parameterestimation, state simulation and mean square optimal procedures areintroduced. The approach described herein is more suitable and robustfor forecasting problems than many existing methods. It can also provideupper and lower bounds for the forecasted state of the system. Further,it applies a nested “two scale hierarchic” quadratic mean-squareoptimization process, whereas existing generalized method of momentsapproaches and their extensions are “single-shot”.

Below, using the role of time-delay processes, the concept of laggedadaptive expectation process, moving average, local finite sequences,local mean and variance, discrete time dynamic sample mean and variancestatistic processes, local conditional and sequences, local sample meanand variance, the DTIDMLSMVSP is developed. A local observation systemis also constructed from nonlinear stochastic functional differentialequations. This can be based on the Itô-Doob stochastic differentialformula and Euler-type numerical scheme in the context of the originalstochastic systems of differential equations and the given data. Inaddition, using the method of moments in the context of lagged adaptiveexpectation process, a procedure is outlined to estimate stateparameters. Using the local lagged adaptive process and the discretetime interconnected dynamic model for statistic process, the idea oftime series data collection schedule synchronization with both numericaland simulation time schedules induces a chain of concepts furtherdescribed below.

The existing GMM-based parameter and state estimation techniques fortesting/selecting continuous-time dynamic models are centered arounddiscretization and model errors in the context of the use of an entiretime-series of data, algebraic manipulations, and econometricspecification for formation of orthogonality condition parameter vectors(OCPV). The existing approaches lead to an overall/single-shot state andparameter estimates, and requires the ergodic stationary condition forconvergence. Furthermore, the existing GMM-based single-shot approachesare not flexible to correctly validate the features of continuous-timedynamic models that are influenced by the state parameter and hereditaryprocesses. In many real-life problems, the past and present dynamicstates influence the future state dynamic. In the formulation of one ofthe components of the LLGMM approach, we incorporate the “past statehistory” via a local lagged adaptive process.

As an introduction to an LLGMM dynamic system according to variousaspects of the embodiments, FIG. 1 illustrates an example computingenvironment 100 for LLGMM dynamic processes. The computing environment100 includes a computing device 110, a data store 120, and an LLGMMdynamic process module 130.

The computing environment 100 can be embodied as one or more computers,computing devices, or computing systems. In certain embodiments, thecomputing environment 100 can include one or more computing devicesarranged, for example, in one or more server or computer banks. Thecomputing device or devices can be located at a single installation siteor distributed among different geographical locations. The computingenvironment 100 can include a plurality of computing devices thattogether embody a hosted computing resource, a grid computing resource,and/or other distributed computing arrangement. One example structure ofthe computing environment 100 is described in greater detail below withreference to FIG. 8.

The data store 120 can be embodied as one or more memories that store(or are capable of storing) and/or embody a discrete time data set 122and state and parameter values 124. In addition, the data store 120 canstore (or is capable of storing) computer readable instructions that,when executed, direct the computing device 110 to perform variousaspects of the LLGMM dynamic processes described herein. In thatcontext, the the data store 120 can store computer readable instructionsthat embody, in part, the LLGMM dynamic process module 130. The discretetime data set 122 can include as past state information of any number ofcontinuous time dynamic processes over any time intervals, as describedin further detail below. Further the state and parameter values 124 caninclude both admissible parameter estimates for a stochastic model of acontinuous time dynamic process and state values of the stochastic modelof the continuous time dynamic process as described in further detailbelow.

The LLGMM dynamic process module 130 includes the dynamic modelgenerator 132, the LLGMM processor 134, the state and parameterestimator 136, and the forecast simulator 138. Briefly, the dynamicmodel generator 132 can be configured to develop a one or morestochastic models of various continuous time dynamic processes. TheLLGMM processor 134 can be configured to generate a DTIDMLSMVSP based onany one of the stochastic models of the continuous time dynamicprocesses developed by the dynamic model generator 132. The state andparameter estimator 136 is configured to calculate a plurality ofadmissible parameter estimates for the stochastic model of thecontinuous time dynamic process using the DTIDMLSMVSP. The state andparameter estimator 136 can be further configured to calculate a statevalue of the stochastic model of the continuous time dynamic process foreach of the plurality of admissible parameter estimates, to gather aplurality of state values of the stochastic model of the continuous timedynamic process. The state and parameter estimator 136 can be furtherconfigured to determine an optimal admissible parameter estimate amongthe plurality of admissible parameter estimates that results in aminimum error among the plurality of state values. Additionally, theforecast simulator 138 can be configured to forecast at least one futurestate value of the stochastic model of the continuous-time dynamicprocess. The functional and operational aspects of the components of theLLGMM dynamic process module 130 are described in greater detail below.

This remainder of this disclosure is organized as follows: in Section 2,using the role of time-delay processes, the concept of lagged adaptiveexpectation process, moving average, local finite sequence, local meanand variance, discrete time dynamic sample mean and variance statisticprocesses, local conditional sequence, and local sample mean andvariance, we develop a general DTIDMLSMVSP. DTIDMLSMVSP is thegeneralization of statistic of random sample drawn from the “static”population. In Section 3, a local observation system is constructed froma nonlinear stochastic functional differential equations. This is basedon the Itô-Doob stochastic differential formula and Euler-type numericalscheme in the context of the original stochastic systems of differentialequations and the given data. In addition, using the method of momentsin the context of lagged adaptive expectation process, a procedure toestimate the state parameters is outlined.

Using the local lagged adaptive process and the discrete timeinterconnected dynamic model for statistic process, the idea of timeseries data collection schedule synchronization with both numerical andsimulation time schedules induces a finite chain of concepts in Section4, namely: (a) local admissible set of lagged sample/data/observationsize, (b) local class of admissible lagged-adapted finite sequence ofconditional sample/data, (c) local admissible sequence of parameterestimates and corresponding admissible sequence of simulated values, (d)ϵ-best sub-optimal admissible subset of set of m_(k)-size localconditional samples at time t_(k) in (a), (e) ϵ-sub-optimallagged-adapted finite sequence of conditional sample/data, and (f) theϵ-best sub optimal parameter estimates and simulated value at time t_(k)for k=1, 2, . . . , N in a systematic way. In addition, the local laggedadaptive process and DTIDMLSMVSP generate a finite chain of discretetime admissible sets/sub-data and corresponding chain described bysimulation algorithm. The usefulness of computational algorithm isillustrated by applying the code not only to four energy commodity datasets, but also to the U.S. Treasury Bill Interest Rate data set and theUSD-EUR Exchange Rate data set in finance for the state and parameterestimation problems. Further, we compare the usage of GARCH (1,1) modelwith the presented DTIDMLSMVSP model. We also compared the DTIDMLSMVSPbased simulated volatility U.S. Treasury Bill Yield Interest rate datawith the simulated work shown in Chan, K. C., Karolyi, G. Andrew,Longstaff, F. A., Sanders, Anthony B., An Empirical Comparison ofAlternative Models of the Short-Term Interest Rate, The Journal ofFinance, Vol. 47., No. 3, 1992, pp. 1209-1227 (“Chan et al”).

In Section 5, the LLGMM is applied to investigate the forecasting andconfidence-interval problems in applied statistics. The presentedresults show the long-run prediction exhibiting a degree of confidence.The use of advancements in electronic communication systems and toolsexhibit that almost everything is dynamic, highly nonlinear,non-stationary and operating under endogenous and exogenous processes.Thus, a multitude of applications of the embodiments described hereinexist. Some extensions include: (a) the development of the DTIDMLSMVSPand (b) the Aggregated Generalized Method of Moments AGMM of the LLGMMmethod are presented in Section 6. In fact, we compare the performanceof DTIDMLSMVSP model with the GARCH(1,1) model and ex post volatility ofChan et al. Further, using the average of locally estimated parametersin the LLGMM, an aggregated generalized method of moment is alsodeveloped and applied to six data sets in Section 6.

In Section 7, a comparative study between the LLGMM and the existingparametric orthogonality condition vector based generalized method ofmoments (OCBGMM) techniques is presented. In Section 8, a comparativestudy between the LLGMM and some existing nonparametric methods is alsopresented. The LLGMM exhibits superior performance to the existing andnewly developed OCBGMM. The LLGMM is problem independent and dynamic. Onthe other hand, the OCBGMM is problem dependent and static. Inappearance, the LLGMM approach seems complicated, but it is userfriendly. It can be operated by a limited theoretical knowledge of theLLGMM. Furthermore, we present several numerical results concerning bothmathematical and applied statistical results showing the comparison ofLLGMM with existing methods.

2. Derivation of Discrete Time Dynamic Model for Sample Mean andVariance Processes

The existing GMM-based parameter and state estimation techniques fortesting/selecting continuous-time dynamic models are centered arounddiscretization and model mispecifications errors in the context of usageof entire time-series data, algebraic manipulations, and econometricspecification for formation of orthogonality condition parameter vectors(OCPV). The existing approaches lead to a single-shot for state andparameter estimates and require the ergodic stationary condition forconvergence. Furthermore, the existing GMM-based single-shot approachesare not flexible to correctly validate the features of continuous-timedynamic models that are influenced by the state parameter and hereditaryprocesses. In many real-life problems, the past and present dynamicstates influence the future state dynamic. In the formulation of one ofthe components of the LLGMM approach, we incorporate the “past statehistory” via local lagged adaptive process.

Further, based on one of the goals of applied mathematical andstatistical research, the embodiments described herein are applicablefor various processes in biological, chemical, engineering, energycommodity markets, financial, medical, and physical and social sciences.Employing the hereditary influence of a systems, the concept of laggedadaptive expectation process, and the idea of moving average, a generalDTIDMLSMVSP is developed with respect to an arbitrary continuous-timestochastic dynamic process. The development of the DTIDMLSMVSP can bemotivated by the state and parameter estimation problems of anycontinuous time nonlinear stochastic dynamic model. Further, the idea ofDTIDMLSMVSP was primarily based on the sample mean and sample varianceideas as statistic for a random sample drawn from a static population inthe descriptive statistics. Using the DTIDMLSMVSP, the problems oflong-term forecasting and interval estimation problems with a highdegree of confidence can be addressed.

For the development of the DTIDMLSMVSP, various definitions andnotations are described herein. Let τ and γ be finite constant timedelays such that 0<γ≤τ. Here, τ characterizes the influence of the pastperformance history of state of dynamic process, and γ describes thereaction or response time delay. In general, these time delays areunknown and random variables. These types of delay play a role indeveloping mathematical models of continuous time and discrete timedynamic processes. Based upon the nature of data collection, it may benecessary to either transform these time delays into positive integersor to design the data collection schedule in relations with thesedelays. For this purpose, the discrete version of time delays of τ and γare defined as

$\begin{matrix}{{r = {\left\lbrack {\frac{\tau}{\Delta\; t_{i}}} \right\rbrack + 1}},{{{and}\mspace{14mu} q} = {\left\lbrack {\frac{\gamma}{\Delta\; t_{i}}} \right\rbrack + 1}},} & (1)\end{matrix}$respectively. For simplicity, we assume that 0<γ<1 (q=1).

Definition 1.

Let x be a continuous time stochastic dynamic process defined on aninterval [−τ, T] into

, for some T>0. For t∈[−τ, T], let

be an increasing sub-sigma algebra of a complete probability space forwhich x(t) is

measurable. Let P be a partition of [−τ, T] defined byP:={t _(i)=−τ+(r+i)Δt}, for i∈I _(−r)(N),  (2)where

${{\Delta\; t} = \frac{\tau + T}{N}},$and I_(i)(k) is defined by I_(i)(k)={j∈

|i≤j≤k}.

Let {x(t_(i))}_(l=−r) ^(N) be a finite sequence corresponding to thestochastic dynamic process x and partition P in Definition 1. Further,x(t_(i)) is

measurable for i∈I_(−r)(N). The definition of forward time shiftoperator F is given by:F ^(i) x(t _(k))=x(t _(k+1)).  (3)Additionally, x(t_(i)) is denoted by x_(i) for i∈I_(−r)(N).

Definition 2.

For q=1 and r≥1, each k∈I₀(N), and each m_(k)∈I₂(r+k−1), a partitionP_(k) of closed interval [t_(k−m) _(k) , t_(k−1)] is called local attime t_(k) and it is defined by

$\begin{matrix}{P_{k}:={t_{k - m_{k}} < t_{k - m_{k} + 1} < \ldots < {t_{k - 1}.}}} & (4)\end{matrix}$P_(k) is referred as the m_(k)-point sub-partition of the partition P in(2) of the closed sub-interval [t_(k−m) _(k) , t_(k−1)] of [−τ,T].

Definition 3.

For each k∈I₀(N) and each m_(k)∈I₂(r+k−1), a local finite sequence at atime t_(k) of the size m_(k) is restriction of {x(t_(l))}_(t=−r) ^(N) toP_(k) in (4), and it is defined by

$\begin{matrix}{S_{m_{k},k}:={\left\{ {F^{i}x_{k - 1}} \right\}_{i = {{- m_{k}} + 1}}^{0}.}} & (5)\end{matrix}$

As m_(k) varies from 2 to k+r−1, the corresponding local sequence S_(m)_(k) _(,k) at t_(k) varies from {x_(i)}_(i=k−2) ^(k−1) to{x_(i)}_(i=−r+1) ^(k−1). As a result of this, the sequence defined in(5) is also called a m_(k)-local moving sequence. Furthermore, theaverage corresponding to the local sequence S_(m) _(k) _(,k) in (5) isdefined by

$\begin{matrix}{{\overset{\_}{S}}_{m_{k},k}:={\frac{1}{m_{k}}{\sum\limits_{i = {{- m_{k}} + 1}}^{0}{F^{i}{x_{k - 1}.}}}}} & (6)\end{matrix}$

The average/mean defined in (6) is also called the m_(k)-localaverage/mean. Further, the m_(k)-local variance corresponding to thelocal sequence S_(m) _(k) _(,k) in (5) is defined by

$\begin{matrix}{S_{m_{k},k}^{2}:=\left\{ {\begin{matrix}{\frac{1}{m_{k}}{\sum\limits_{i = {{- m_{k}} + 1}}^{0}\;\left( {{F^{i}x_{k - 1}} - {\frac{1}{m_{k}}{\sum\limits_{j = {{- m_{k}} + 1}}^{0}{F^{j}x_{k - 1}}}}} \right)^{2}}} & {{for}\mspace{14mu}{small}\mspace{14mu} m_{k}} \\{\frac{1}{m_{k} - 1}{\sum\limits_{i = {{- m_{k}} + 1}}^{0}\;\left( {{F^{i}x_{k - 1}} - {\frac{1}{m_{k}}{\sum\limits_{j = {{- m_{k}} + 1}}^{0}{F^{j}x_{k - 1}}}}} \right)^{2}}} & {{for}\mspace{14mu}{large}\mspace{14mu} m_{k}}\end{matrix}.} \right.} & (7)\end{matrix}$

Definition 4.

For each fixed k∈I₀(N), and any m_(k)∈I₂ (k+r−1), the sequence {S_(i,k)}_(i=k−m) _(k) ^(k−1) is called a m_(k)-local moving average/meanprocess at t_(k). In other words, the LLGMM dynamic process includes,for each m_(k)-local moving sequence, calculating an m_(k)-local averageto generate an m_(k)-moving average process (e.g., reference numeral 310in FIG. 3). Further, the sequence {s_(i,k) ²}_(i=k−m) _(k) ^(k−1) iscalled a m_(k)-local moving variance process at t_(k). That is, for eachm_(k)-local moving sequence, the process includes calculating anm_(k)-local variance to generate an m_(k)-local moving variance process(e.g., reference numeral 312 in FIG. 3).

Definition 5.

Let {x(t_(i))}_(l=−r) ^(N) be a random sample of continuous timestochastic dynamic process collected at partition P in (2). The localsample average/mean in (6) and local sample variance in (7) are calleddiscrete time dynamic processes of sample mean and sample variancestatistics.

Definition 6.

Let {x(t_(i))}_(i=−r) ^(N) be a random sample of continuous timestochastic dynamic process collected at partition P in (2). Them_(k)-local moving average and variance defined in (6) and (7) arecalled the m_(k)-local moving sample average/mean and local movingsample variance at time t_(k), respectively. Further, m_(k)-local sampleaverage and m_(k)-local sample variance are referred to as local samplemean and local sample variance statistics for the local mean andvariance of the continuous time stochastic dynamic process at timet_(k), respectively. S _(m) _(k) and s_(m) _(k) ² are called samplestatistic time series processes.

Definition 7.

Let {

[x(t_(i))|

]}_(i=−r+1) ^(N) be a conditional random sample of continuous timestochastic dynamic process with respect to sub-σ-algebra

, t_(i)∈P in (2). The m_(k)-local conditional moving average andvariance defined in the context of (6) and (7) are called them_(k)-local conditional moving sample average/mean and local conditionalmoving sample variance, respectively.

The concept of sample statistic time-series/process extends the conceptof random sample statistic for static dynamic populations in a naturaland unified way. Employing Definition 7, we introduce the DTIDMLSMVSP.As described in detail below, this discrete time algorithm/model playsan important role in state and parameter estimation problems fornonlinear and non-stationary continuous-time stochastic differential anddifference equations. Further, it provides feedback for bothcontinuous-time dynamic model and corresponding discrete time statisticdynamic model for modifications and updates under the influence ofexogenous and endogenous varying forces or conditions in a systematicand unified way. It is also clear that the discrete time algorithm easesthe updates in the time-series statistic. Now, a change in S _(m) _(k)_(,k) and s_(m) _(k) _(,k) ² with respect to change in time t_(k) can bestated.

Lemma 1. (DTIDMLSMVSP).

Let {

[x(t_(i))|

]}_(i=−r+1) ^(N) be a conditional random sample of continuous timestochastic dynamic process with respect to sub-σ-algebra

, t_(i) belong to partition P in. Let S _(m) _(k) _(,k) and s_(m) _(k)_(,k) ² be m_(k)-local conditional sample average and local conditionalsample variance at t_(k) for each k∈I₀(N). Using these inputs (e.g.,reference numeral 314 in FIG. 3), an example DTIDMLSMVSP can bedescribed by

$\begin{matrix}\left\{ \begin{matrix}{\overset{\_}{S}}_{m_{k - p + 1},{k - p + 1}} & = & \begin{matrix}{{\frac{m_{k - p}}{m_{k - p + 1}}{\overset{\_}{S}}_{m_{k - p + 1},{k - p}}} +} \\{\eta_{m_{k - p},{k - p}},{{\overset{\_}{S}}_{m_{0},0} = {\overset{\_}{S}}_{0}}}\end{matrix} \\S_{m_{k},k}^{2} & = & \left\{ \begin{matrix}\begin{matrix}{\frac{m_{k - 1}}{m_{k}}\left\lbrack {\sum\limits_{i = 1}^{p}\;\left\lbrack \frac{m_{k - i}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}} \right\rbrack} \right.} \\{s_{m_{k - i},{k - i}}^{2} +} \\{\left\lbrack {\frac{m_{k - p}}{\prod\limits_{j = 0}^{p - 1}\; m_{k - j}}{\overset{\_}{S}}_{m_{k - p},{k - p}}^{2}} \right\rbrack +} \\{ɛ_{m_{k - 1},{k - 1}},}\end{matrix} & \begin{matrix}{{{for}\mspace{14mu}{small}\mspace{14mu} m_{k}},} \\{m_{k - 1} \leq m_{k}}\end{matrix} \\\begin{matrix}{\sum\limits_{i = 1}^{p}\;\left\lbrack \frac{m_{k - i} - 1}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}} \right\rbrack} \\{s_{m_{k - i},{k - i}}^{2} +} \\{{\frac{m_{k - p}}{\prod\limits_{j = 0}^{p - 1}\; m_{k - j}}{\overset{\_}{S}}_{m_{k - p},{k - p}}^{2}} +} \\{\epsilon_{m_{k - 1},{k - 1}},}\end{matrix} & \begin{matrix}{{{for}\mspace{14mu}{large}\mspace{14mu} m_{k}},} \\{m_{k - 1} \leq m_{k}}\end{matrix} \\{{s_{m_{i},i}^{2} = s_{i}^{2}},{i \in {I_{- p}(0)}},} & {{initial}\mspace{14mu}{conditions}}\end{matrix} \right.\end{matrix} \right. & (8)\end{matrix}$where

$\begin{matrix}\left\{ \begin{matrix}{\eta_{m_{k - p},{k - p}} = \begin{matrix}{\frac{1}{m_{k - p + 1}}\left\lbrack {{\sum\limits_{i = {{- m_{k - p + 1}} + 1}}^{{- m_{k - p}} + 1}\;{F^{i}x_{k - p}}} -} \right.} \\{\left. {{F^{{- m_{k - p}} + 1}x_{k - p}} - {F^{- m_{k - p}}x_{k - p}} + {F^{0}x_{k - p}}} \right\rbrack,}\end{matrix}} \\{ɛ_{m_{k - 1},{k - 1}} = \begin{matrix}{\frac{m_{k} - 1}{m_{k}}\left\lbrack {{\sum\limits_{i = 1}^{p}\;\frac{\left( {F^{{- i} + 1}x_{k - 1}} \right)^{2}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}}} -} \right.} \\{{\sum\limits_{i = 1}^{p}\;\frac{\left( {F^{{- i} + 1 - m_{k - i}}x_{k - 1}} \right)^{2}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}}} -} \\{\left. {\sum\limits_{i = 1}^{p}\;\frac{\left( {F^{{- i} + 2 - m_{k - i}}x_{k - 1}} \right)^{2}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}}} \right\rbrack +} \\{\frac{m_{k} - 1}{m_{k}}\left\lbrack {{\sum\limits_{i = 1}^{p}\left\lbrack \frac{\sum\limits_{l = {{- i} + 2 - m_{k - i + 1}}}^{{- i} + 2 - m_{k - i}}\;\left( {F^{l}x_{k - 1}} \right)^{2}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}} \right\rbrack} +} \right.} \\{\left. {\sum\limits_{i = 1}^{p}\left\lbrack \frac{\sum\limits_{\underset{l \neq s}{l,{s = {{- i} + 2 - m_{k - i + 1}}}}}^{{- i} + 1}\;{F^{l}x_{k - 1}F^{s}x_{k - 1}}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}} \right\rbrack} \right\rbrack -} \\{{\frac{1}{m_{k}}{\sum\limits_{\underset{l \neq s}{l,{s = {{- m_{k}} + 1}}}}^{0}{F^{l}x_{k - 1}F^{s}x_{k - 1}}}},}\end{matrix}}\end{matrix} \right. & (9)\end{matrix}$

$\left\{ \begin{matrix}{\epsilon_{m_{k - 1},{k - 1}} = {{\sum\limits_{i = 1}^{p}\;\frac{\left( {F^{{- i} + 1}x_{k - 1}} \right)^{2}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}}} - {\sum\limits_{i = 1}^{p}\;\frac{\left( {F^{{- i} + 1 - m_{k - i}}x_{k - 1}} \right)^{2}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}}} -}} \\{{\sum\limits_{i = 1}^{p}\;\frac{\left( {F^{{- i} + 2 - m_{k - i}}x_{k - 1}} \right)^{2}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}}} + {\sum\limits_{i = 1}^{p}\left\lbrack \frac{\sum\limits_{l = {{- i} + 2 - m_{k - i + 1}}}^{{- i} + 2 - m_{k - i}}\;\left( {F^{l}x_{k - 1}} \right)^{2}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}} \right\rbrack} +} \\{{\sum\limits_{i = 1}^{p}\left\lbrack \frac{\sum\limits_{\underset{l \neq s}{l,{s = {{- i} + 2 - m_{k - i + 1}}}}}^{{- i} + 1}\;{F^{l}x_{k - 1}F^{s}x_{k - 1}}}{\prod\limits_{j = 0}^{i - 1}\; m_{k - j}} \right\rbrack} - {\frac{1}{m_{k} - 1}{\sum\limits_{\underset{l \neq s}{l,{s = {{- m_{k}} + 1}}}}^{0}{F^{l}x_{k - 1}F^{s}x_{k - 1}}}}}\end{matrix} \right.$

Remark 1.

The interconnected dynamic statistic system (8) can be re-written as theone-step Gauss-Sidel dynamic system of iterative process described by

$\begin{matrix}{\mspace{79mu}{{{X\left( {k;p} \right)} = {{{A\left( {k,{{X\left( {{k - 1};p} \right)};p}} \right)}{X\left( {{k - 1};p} \right)}} + {e\left( {k;p} \right)}}},\mspace{20mu}{{{where}\mspace{14mu}{X\left( {k;p} \right)}} = \begin{pmatrix}{X_{1}\left( {k;p} \right)} \\{X_{2}\left( {k;p} \right)}\end{pmatrix}},\mspace{20mu}{{X_{1}\left( {k;p} \right)} = {\overset{\_}{S}}_{{m_{k - p + 1}k} - p + 1}},{{X_{2}(k)} = \begin{pmatrix}S_{m_{k - p + 1},{k - p + 1}}^{2} \\S_{m_{k - p + 2},{k - p + 2}}^{2} \\\vdots \\S_{m_{k - 1},{k - 1}}^{2} \\S_{m_{k},k}^{2}\end{pmatrix}},{{A\left( {k,{{X\left( {{k - 1};p} \right)};p}} \right)} = \begin{pmatrix}{A_{11}\left( {k;p} \right)} & {A_{12}\left( {k;p} \right)} \\{A_{21}\left( {k,{{X\left( {{k - 1};p} \right)};p}} \right)} & {A_{22}\left( {k;p} \right)}\end{pmatrix}},\mspace{20mu}{{A_{11}\left( {k;p} \right)} = \frac{m_{k - p}}{m_{k} - p + 1}},\mspace{20mu}{{A_{12}\left( {k;p} \right)} = \begin{pmatrix}0 & 0 & \ldots & 0\end{pmatrix}},}} & (10)\end{matrix}$

${A_{21}\left( {k;p} \right)} = \left\{ {\begin{matrix}{\begin{pmatrix}0 \\0 \\\vdots \\0 \\{\frac{\left( {m_{k} - 1} \right)m_{k - p}}{m_{k}{\prod\limits_{j = 0}^{p - 1}\; m_{k - j}}}{\overset{\_}{S}}_{m_{k - p},{k - p}}}\end{pmatrix},} & {{for}\mspace{14mu}{small}\mspace{14mu} m_{k}} \\{\begin{pmatrix}0 \\0 \\\vdots \\0 \\{\frac{m_{k - p}}{\prod\limits_{j = 0}^{p - 1}\; m_{k - j}}{\overset{\_}{S}}_{m_{k - p},{k - p}}}\end{pmatrix},} & {{{for}\mspace{14mu}{large}\mspace{14mu} m_{k}},}\end{matrix},} \right.$

${{{A_{22}\left( {k;p} \right)} = \begin{pmatrix}0 & 1 & 0 & 0 & \ldots & 0 \\0 & 0 & 1 & 0 & \ldots & 0 \\\vdots & 0 & 0 & 0 & \ddots & \vdots \\0 & \ldots & 0 & 0 & 0 & 1 \\\frac{\left( {m_{k} - 1} \right)m_{k - p}}{m_{k}{\prod\limits_{j = 0}^{p - 1}\; m_{k - j}}} & \frac{\left( {m_{k} - 1} \right)m_{k - p + 1}}{m_{k}{\prod\limits_{j = 0}^{p - 2}\; m_{k - j}}} & \ldots & \frac{\left( {m_{k} - 1} \right)m_{k - p + i - 1}}{m_{k}{\prod\limits_{j = 0}^{p - i}\; m_{k - j}}} & \ldots & \frac{\left( {m_{k} - 1} \right)m_{k - 1}}{m_{k}^{2}}\end{pmatrix}},{{for}\mspace{14mu}{small}\mspace{14mu} m_{k}},{{and}\mspace{14mu}\begin{pmatrix}0 & 1 & 0 & 0 & \ldots & 0 \\0 & 0 & 1 & 0 & \ldots & 0 \\\vdots & 0 & 0 & 0 & \ddots & \vdots \\0 & \ldots & 0 & 0 & 0 & 1 \\\frac{m_{k - p} - 1}{\prod\limits_{j = 0}^{p - 1}\; m_{k - j}} & \frac{m_{k - p + 1} - 1}{\prod\limits_{j = 0}^{p - 2}\; m_{k - j}} & \ldots & \frac{m_{k - p + i - 1} - 1}{\prod\limits_{j = 0}^{p - i}\; m_{k - j}} & \ldots & \frac{m_{k - 1} - 1}{m_{k}^{2}}\end{pmatrix}},{{for}\mspace{14mu}{large}\mspace{14mu} m_{k}}}\mspace{14mu}$

${{e\left( {k;p} \right)} = \begin{pmatrix}{e_{1}\left( {k;p} \right)} \\{e_{2}\left( {k;p} \right)}\end{pmatrix}},{{e_{1}\left( {k;p} \right)} = \eta_{m_{k - p},{k - p}}},{{e_{2}\left( {k;p} \right)} = \begin{pmatrix}0 \\0 \\\vdots \\\epsilon_{m_{k - 1},{k - 1}}^{*}\end{pmatrix}},{\epsilon_{m_{k - 1},{k - 1}}^{*} = \left\{ \begin{matrix}{\epsilon_{m_{k - 1},{k - 1}},} & {{for}\mspace{14mu}{small}\mspace{14mu} m_{k}} \\{\epsilon_{m_{k - 1},{k - 1}},} & {{for}\mspace{14mu}{large}\mspace{14mu} m_{k}}\end{matrix} \right.}$

Remark 2.

For each k∈I₀(N), p=2, and small m_(k), the inter-connected system (8)reduces to the following special caseX(k;2)=A(k,

(k−1;2);2)X(k−1;2)+e(k;2),  (11)where X(k; 2), A(k; 2) and e(k; 2) are defined by

${{X\left( {k;2} \right)} = \begin{pmatrix}{X_{1}\left( {k;2} \right)} \\{X_{2}\left( {k;2} \right)}\end{pmatrix}},$X₁(k;2)=S _(m) _(k−1) _(,k−1),

${{X_{2}\left( {k;2} \right)} = \begin{pmatrix}s_{m_{k - 1},{k - 1}}^{2} \\s_{m_{k},k}^{2}\end{pmatrix}},{{A\left( {k;2} \right)} = \begin{pmatrix}{A_{11}\left( {k;2} \right)} & {A_{12}\left( {k;2} \right)} \\{A_{21}\left( {k;2} \right)} & {A_{22}\left( {k;2} \right)}\end{pmatrix}},{{A_{11}\left( {k;2} \right)} = \frac{m_{k - 2}}{m_{k} - 1}},{{A_{12}\left( {k;2} \right)} = \begin{pmatrix}0 & 0\end{pmatrix}},{{A_{21}\left( {k;2} \right)} = \begin{pmatrix}0 \\{\frac{\left( {m_{k} - 1} \right)m_{k - 2}}{m_{k}^{2}m_{k - 1}}{\overset{\_}{S}}_{m_{k - 2},{k - 2}}}\end{pmatrix}},{{A_{22}\left( {k;2} \right)} = \begin{pmatrix}0 & 1 \\\frac{\left( {m_{k} - 1} \right)m_{k - 2}}{m_{k}^{2}m_{k - 1}} & \frac{\left( {m_{k} - 1} \right)m_{k - 1}}{m_{k}^{2}}\end{pmatrix}},{{{e\left( {k;2} \right)} = \begin{pmatrix}{e_{1}\left( {k;2} \right)} \\{e_{2}\left( {k;2} \right)}\end{pmatrix}};{{e_{1}\left( {k;2} \right)} = \eta_{m_{k - 2},{k - 2}}}},{{e_{2}\left( {k;2} \right)} = \begin{pmatrix}0 \\ɛ_{m_{k - 1},{k - 1}}\end{pmatrix}}$

$\quad\left\{ \begin{matrix}\eta_{m_{k - 2},{k - 2}} & = & \begin{matrix}{\frac{1}{m_{k}}\left\lbrack {{\sum\limits_{i = {{- m_{k - 1}} + 1}}^{{- m_{k - 2}} + 1}\;{F^{i}x_{k - 2}}} -} \right.} \\{\left. {{F^{{- m_{k - 2}} + 1}x_{k - 2}} - {F^{- m_{k - 2}}x_{k - 2}} + {F^{0}x_{k - 2}}} \right\rbrack,}\end{matrix} \\ɛ_{m_{k - 1},{k - 1}} & = & \begin{matrix}{\frac{m_{k} - 1}{m_{k}}\left\lbrack {\frac{\begin{matrix}{\left( {F^{0}x_{k - 1}} \right)^{2} - \left( {F^{- m_{k - 1}}x_{k - 1}} \right)^{2} -} \\\left( {F^{1 - m_{k - 1}}x_{k - 1}} \right)^{2}\end{matrix}}{m_{k}} +} \right.} \\{\left. \frac{\left( {F^{- 1}x_{k - 1}} \right)^{2} - \left( {F^{{- 1} - m_{k - 2}}x_{k - 1}} \right)^{2} - \left( {F^{- m_{k - 2}}x_{k - 1}} \right)^{2}}{m_{k}m_{k - 1}} \right\rbrack +} \\{\frac{m_{k} - 1}{m_{k}}\left\lbrack {\frac{\sum\limits_{i = {- m_{k - 1}}}^{- m_{k - 2}}\;\left( {F^{i}x_{k - 1}} \right)^{2}}{m_{k}m_{k - 1}} + \frac{\sum\limits_{\underset{i \neq j}{i,{j = {- m_{k - 1}}}}}^{- 1}\;{F^{i}x_{k - 1}F^{j}x_{k - 1}}}{m_{k}m_{k - 1}} +} \right.} \\{\left. \frac{\sum\limits_{i = {1 - m_{k}}}^{1 - m_{k - 1}}\;\left( {F^{i}x_{k - 1}} \right)^{2}}{m_{k}} \right\rbrack - {\frac{\sum\limits_{\underset{i \neq j}{i,{j = {1 - m_{k}}}}}^{0}\;{F^{i}x_{k - 1}F^{j}x_{k - 1}}}{m_{k}^{2}}.}}\end{matrix}\end{matrix} \right.$

Remark 3.

Define

${\varphi_{1} = {\frac{m_{k} - 1}{m_{k}}\frac{m_{k - 1}}{m_{k}}}},{\varphi_{2} = {\frac{m_{k} - 1}{m_{k}}\frac{m_{k - 2}}{m_{k}m_{k - 1}}}},{{{and}\mspace{14mu}\varphi_{3}} = {\frac{m_{k - 2}}{m_{k - 1}}.}}$For small m_(k), m_(k−1)≤m_(k), ∀k, we have φ₁<1, φ₂<1, and φ₃≤1. From0<φ_(i), i=1, 2, 3, and the fact that

${{\varphi_{1} + \varphi_{2}} = {{\frac{m_{k} - 1}{m_{k}^{2}}\left\lbrack {m_{k - 1} + \frac{m_{k - 2}}{m_{k - 1}}} \right\rbrack} \leq {\frac{m_{k} - 1}{m_{k}^{2}}\left\lbrack {m_{k - 1} + 1} \right\rbrack} \leq \frac{m_{k}^{2} - 1}{m_{k}^{2}} < 1}},$the stability of the trivial solution (e.g., X(k; 2)=0) of thehomogeneous system corresponding to (10) follows. Further, under theabove stated conditions, the convergence of solutions of (10) alsofollows.

Remark 4.

From Remark 2, the local sample variance statistics at time t_(k)depends on the state of the m_(k−1) and m_(k−2)-local sample variancestatistics at time t_(k−1) and t_(k−2), respectively, and them_(k−2)-local sample mean statistics at time t_(k−2).

Remark 5.

Aspects of the role and scope of the DTIDMLSMVSP can be summarized.First, the DTIDMLSMVSP is the second component of the LLGMM approach.The DTIDMLSMVSP is valid for a transformation of data. It isgeneralization of a “statistic” of a random sample drawn from “static”population problems. Further, Lemma 1 provides iterative scheme forupdating statistic coefficients in the local systems ofmoment/observation equations in the LLGMM approach. This accelerates thespeed of computation. The DTIDMLSMVSP does not require any type ofstationary condition. The DTIDMLSMVSP plays a significant role in thelocal discretization and model validation errors. Finally, the approachto the DTIDMLSMVSP is more suitable for forecasting problems, as furtheremphasized in the subsequent sections.

Remark 6.

The usefulness of the DTIDMLSMVSP arises in estimation of volatilityprocess of a stochastic differential or difference equations. This modelprovides an alternative approach to the GARCH(p,q) model. Below, them_(k)-local sample variance statistics are compared with the GARCH(p,q)model to show that the m_(k)-local sample variance statistics give abetter forecast than the GARCH(p,q) model.

3. Theoretical Parametric Estimation Procedure

In this section, a foundation based on a mathematically rigoroustheoretical state and parameter estimation procedure is formulated for avery general continuous-time nonlinear and non-stationary stochasticdynamic model described by a system stochastic differential equations.This work is not only motivated by the continuous-time dynamic modelvalidation problem in the context of real data energy commodities, butalso motivated by any continuous-time nonlinear and non-stationarystochastic dynamic model validation problems in biological, chemical,engineering, financial, medical, physical and social sciences, amongothers. This is because of the fact that the development of the existingOrthogonality Condition Based GMM (OCBGMM) procedure is primarilycomposed of the following five components: (1) testing/selectingcontinuous-time stochastic models for a particular dynamic processdescribed by one or more stochastic differential equations, (2) usingeither a Euler-type discretization scheme, a discrete time econometricspecification, or other discretization scheme regarding the stochasticdifferential equation specified in (1), (3) forming an orthogonalitycondition parameter vector (OCPV) using algebraic manipulation, (4)using (2), (3) and the entire time series data set, finding a system ofmoment equations for the OCBGMM, and (5) single-shot parameter and stateestimates using positive-definite quadratic form. The existing OCBGMMlacks the usage of Itô-Doob calculus, properties of stochasticdifferential equations, and a connection with econometric baseddiscretization schemes, the orthogonality conditional vector, and thequadratic form.

In this section, an attempt is made to eliminate the drawbacks,operational limitations, and the lack of connectivity and limited scopeof the OCBGMM. This is achieved by utilizing (i) historical role playedby hereditary process in dynamic modeling, (ii) Itô-Doob calculus, (iii)the fundamental properties of stochastic system of differentialequations, (iv) the lagged adaptive process, (v) the discrete timeinterconnected dynamics of local sample mean and variances statisticprocesses model in Section 2 (Lemma 1), (vi) the Euler-type numericalschemes for both stochastic differential equations generated from theoriginal stochastic systems of differential equations and the originalstochastic systems of differential equations, (vii) systems ofmoments/observation equations, and (viii) local observation/measurementssystems in the context of real world data.

Starting in this section, parts of the the LLGMM dynamic process 200shown in FIG. 2 are also described. At reference numeral 202, theprocess 200 includes obtaining a discrete time data set as past stateinformation of a continuous time dynamic process over a time interval,such as the [−τ,T] described herein. The discrete time data set can bestored in the data store 120 as the discrete time data set 122. Further,at reference numeral 204, the process 200 includes developing astochastic model of a continuous time dynamic process.

As one example of a stochastic model of a continuous time dynamicprocess, a general system of stochastic differential equations under theinfluence of hereditary effects in both the drift and diffusioncoefficients is described bydy=f(t,y _(t))dt+σ(t,y _(t))dW(t),y _(t) ₀ =φ₀,  (12)where, y_(t)(θ)=y(t+θ), θ∈[−τ,0], f, σ: [0, T]×C→

are Lipschitz continuous bounded functionals,

is the Banach space of continuous functions defined on [−τ,0] into

equipped with the supremum norm, W(t) is standard Wiener process definedon a complete filtered probability space (Ω,

), φ₀∈C, y₀(t₀+θ) is

)_(t) ₀ measurable, the filtration function (

)_(t≥0) is right-continuous, each

_(t) with t≥t₀ contains all

-null events in F, and the solution process y(t₀,φ₀)(t) is adapted andnon-anticipating with respect to (

)_(t≥0).

3.1 Transformation of System of Stochastic Differential Equations (12)

At reference numeral 206, the process 200 includes generating aDTIDMLSMVSP based on the stochastic model of the continuous time dynamicprocess. As part of the conceptual aspects of generating theDTIDMLSMVSP, at reference numeral 206, the process 200 can includetransforming the stochastic model of the continuous time dynamic processinto a stochastic model of a discrete time dynamic process utilizing adiscretization scheme. For example, let V∈C[[−τ, ∞]×

]. Its partial derivatives V_(t),

$V_{t},\frac{\partial V}{\partial y},\frac{\partial^{2}V}{\partial y^{2}}$exist and are continuous. The Itô-Doob stochastic differential formulacan be applied to V to obtaindV(t,y)=LV(t,y,y _(t))dt+V _(y)(t,y)σ(t,y _(t))dW(t),  (13)where the L operator is defined by

$\begin{matrix}\left\{ \begin{matrix}{{LV}\left( {t,y,y_{t}} \right)} & = & {{V_{t}\left( {t,y} \right)} + {{V_{y}\left( {t,y} \right)}{f\left( {t,y_{t}} \right)}} + {\frac{1}{2}{{tr}\left( {{V_{yy}\left( {t,y} \right)}{b\left( {t,y_{t}} \right)}} \right)}}} \\{b\left( {t,y_{t}} \right)} & = & {{\sigma\left( {t,y_{t}} \right)}{{\sigma^{T}\left( {t,y_{t}} \right)}.}}\end{matrix} \right. & (14)\end{matrix}$

3.2 Euler-Type Discretization Scheme for (12) and (13)

For (12) and (13), the Euler-type discretization scheme can be presentedas

$\begin{matrix}\left\{ {\begin{matrix}{\Delta\; y_{i}} & = & \begin{matrix}{{{f\left( {t_{i - 1},y_{t_{i - 1}}} \right)}\Delta\; t_{i}} +} \\{{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)\Delta\; W_{i - 1}},{i \in {I_{1}(N)}}}\end{matrix} \\{\Delta\;{V\left( {t_{i},{y\left( t_{i} \right)}} \right)}} & = & \begin{matrix}{{{{LV}\left( {t_{i - 1},{y\left( t_{i} \right)},y_{t_{i - 1}}} \right)}\Delta\; t_{i}} +} \\{{V_{y}\left( {t_{i - 1},{y\left( t_{i - 1} \right)}} \right)}{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)}\Delta\;{W\left( t_{i} \right)}}\end{matrix}\end{matrix},} \right. & (15)\end{matrix}$and

_(t) _(i−1) ≡

_(i−1) can be defined as the filtration process up to time t_(i−1).

3.3 Formation of Generalized Moment Equations from (15)

As another part of the conceptual aspects of generating the DTIDMLSMVSP,at reference numeral 206, the process 200 can also include developing asystem of generalized method of moments equations from the stochasticmodel of the discrete time dynamic process. For example, with regard tothe continuous time dynamic system (12) and its transformed system (13),the more general moments of Δy(t_(i)) are:

$\quad\left\{ {\begin{matrix}{E\left\lbrack {{\Delta\;{y\left( t_{i} \right)}}❘} \right.} & = & {{{f\left( {t_{i - 1},y_{t_{i - 1}}} \right)}\Delta\; t_{i}},} \\\begin{matrix}{E\left\lbrack \left( {{\Delta\;{y\left( t_{i} \right)}} - {E\left\lbrack {{\Delta\;{y\left( t_{i} \right)}}❘} \right.}} \right. \right.} \\\left( {{{\Delta\;{y\left( t_{i} \right)}} - {E\left\lbrack {{\Delta\;{y\left( t_{i} \right)}}❘} \right)}^{T}}❘\mathcal{F}_{i - 1}} \right\rbrack\end{matrix} & = & \begin{matrix}{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)} \\{{{\sigma^{T}\left( {t_{i - 1},y_{t_{i - 1}}} \right)}\Delta\; t_{i}},}\end{matrix} \\{E\left\lbrack {{\Delta\;{V\left( {t_{i},{y\left( t_{i} \right)}} \right)}}❘} \right.} & = & {{{LV}\left( {t_{i - 1},{y\left( t_{i} \right)},y_{t_{i - 1}}} \right)}\Delta\; t_{i}} \\\begin{matrix}{E\left\lbrack \left( {{\Delta\; V\left( {t_{i},{y\left( t_{i} \right)}} \right)} - {E\left\lbrack {\Delta\;{V\left( {t_{i},{y\left( t_{i} \right)}} \right)}} \right.}} \right. \right.} \\{\left. \left. {❘\mathcal{F}_{i - 1}} \right\rbrack \right)\left( {{\Delta\;{V\left( {t_{i},{y\left( t_{i} \right)}} \right)}} -} \right.} \\\left. {{E\left\lbrack {{\Delta\;{V\left( {t_{i},{y\left( t_{i} \right)}} \right)}}❘} \right)}^{T}❘\mathcal{F}_{i - 1}} \right\rbrack\end{matrix} & = & {B\left( {t_{i - 1},{y\left( t_{i - 1} \right)},y_{t_{i - 1}}} \right)}\end{matrix},} \right.$whereB(t_(i−1),y(t_(i−1)),y_(t−1))=V_(y)(t_(i−1),y(t_(i−1)))b(t_(i−1),y_(t−1))V_(y)(t_(i−1),y(t_(i−1)))^(T)Δ^(t),and T stands for the transpose of the matrix.

3.4 Basis for Local Lagged Adaptive Discrete Time Expectation Process

From (15) and (16),

$\begin{matrix}\left\{ {\begin{matrix}{\Delta\; y_{i}} & = & \begin{matrix}{{E\left\lbrack {{\Delta\;{y\left( t_{i} \right)}}❘\mathcal{F}_{i - 1}} \right\rbrack} +} \\{{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)\Delta\; W_{i - 1}},{i \in {I_{1}(N)}}}\end{matrix} \\{\Delta\;{V\left( {t_{i},{y\left( t_{i} \right)}} \right)}} & = & \begin{matrix}{{E\left\lbrack {{\Delta\;{V\left( {t_{i},{y\left( t_{i} \right)}} \right)}}❘\mathcal{F}_{i - 1}} \right\rbrack} +} \\{{V_{y}\left( {t_{i - 1},{y\left( t_{i - 1} \right)}} \right)}{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)}\Delta\;{W\left( t_{i} \right)}}\end{matrix}\end{matrix}.} \right. & (17)\end{matrix}$This provides the basis for the development of the concept of laggedadaptive expectation with respect to continuous time stochastic dynamicsystems (12) and (13). This also leads to a formulation of m_(k)-localgeneralized method of moments at t_(k).

Remark 7.

(Block Orthogonality Condition Vector for (12) and (13)). From (17), onecan define a block vector of orthogonality condition as

$\begin{matrix}{{H\left( {t_{i - 1},{y\left( t_{i} \right)},{y\left( t_{i - 1} \right)}} \right)} = {\begin{pmatrix}{{\Delta\;{y\left( t_{i} \right)}} - {{f\left( {t_{i - 1},{y\left( t_{i - 1} \right)}} \right)}\Delta\; t_{i}}} \\{{\Delta\;{V\left( {t_{i - 1},{y\left( t_{i} \right)}} \right)}} - {{{LV}\left( {t_{i - 1},{y\left( {t_{i - 1},y_{t_{i - 1}}} \right)}} \right)}\Delta\; t_{i}}}\end{pmatrix}.}} & (18)\end{matrix}$Further, unlike the orthogonality condition vector defined in theliterature, the definition of the block vector of orthogonalitycondition (18) is based on the discretization scheme associated withnonlinear and non-stationary continuous-time stochastic system ofdifferential equations (12) and (13) and the Itô-Doob stochasticdifferential calculus.

Example 1

For V(t,y) in (13) defined by

$\begin{matrix}{\mspace{79mu}{{{V\left( {t,y} \right)} = {{y}_{p}^{p} = {\sum\limits_{j = 1}^{n}\;{y^{j}}^{p}}}},{{dV} = {{\left\lbrack {{p{\sum\limits_{j = 1}^{n}\;{{y^{j}}^{p - 1}{{sgn}\left( y^{j} \right)}{f\left( {t,y_{t}^{j}} \right)}}}} + {\frac{p\left( {p - 1} \right)}{2}{y^{j}}^{p - 2}{\sigma\left( {t,y_{t}^{j}} \right)}}} \right\rbrack{dt}} + {p{\sum\limits_{j = 1}^{n}\;{{y^{j}}^{p - 1}{{sgn}\left( y^{j} \right)}{\sigma\left( {t,y_{t}^{j}} \right)}{{dW}^{j}.}}}}}}}} & (19)\end{matrix}$

Hence, the discretized form of (19) is given by

$\begin{matrix}{{\Delta\; V_{i}} = {{\left\lbrack {{p{\sum\limits_{j = 1}^{n}\;{{y_{i - 1}^{j}}^{p - 1}{{sgn}\left( y_{i - 1}^{j} \right)}{f\left( {t_{i - 1},y_{t_{i - 1}}^{j}} \right)}}}} + {\frac{p\left( {p - 1} \right)}{2}{y_{i - 1}^{j}}^{p - 2}{\sigma\left( {t_{i - 1},y_{t_{i - 1}}^{j}} \right)}}} \right\rbrack{dt}} + {p{\sum\limits_{j = 1}^{n}\;{{y_{i - 1}^{j}}^{p - 1}{{sgn}\left( y_{i - 1}^{j} \right)}{\sigma\left( {t_{i - 1},y_{t_{i - 1}}^{j}} \right)}{{dW}_{i}^{j}.}}}}}} & (20)\end{matrix}$In this special case, (17) reduces to

$\begin{matrix}\left\{ {\begin{matrix}{\Delta\; y_{i}} & = & \begin{matrix}{{E\left\lbrack {{\Delta\;{y\left( t_{i} \right)}}❘\mathcal{F}_{i - 1}} \right\rbrack} +} \\{{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)\Delta\; W_{i - 1}},{i \in {I_{1}(N)}}}\end{matrix} \\{\Delta\left( \;{\sum\limits_{j = 1}^{n}{y_{i}^{j}}^{p}} \right)} & = & \begin{matrix}{{E\left\lbrack {{\Delta\left( \;{\sum\limits_{j = 1}^{n}{y_{i}^{j}}^{p}} \right)}❘\mathcal{F}_{i - 1}} \right\rbrack} +} \\{p{\sum\limits_{j = 1}^{n}{{y_{i - 1}^{j}}^{p - 1}{{sgn}\left( y_{i - 1}^{j} \right)}}}} \\{\sigma\left( {t_{i - 1},y_{t_{i - 1}}^{j}} \right){dW}_{i}^{j}}\end{matrix}\end{matrix}.} \right. & (21)\end{matrix}$

Example 2

We consider a multivariate AR(1) model as another example to exhibit theparameter and state estimation problem. The AR(1) model is of thefollowing typex _(t) =a _(t−1) x _(t−1)+σ_(t−1) e _(t) ,x(0)=x ₀, for t=0,1,2, . . .,t, . . . ,N,  (22)where x_(t), x₀∈

, e_(t)∈

is

a measurable normalized discrete time Gaussian process, and a_(t−1) andσ_(t−1) are n×n and n×m discrete time varying matrix functions,respectively. Here

$\begin{matrix}{\begin{pmatrix}{E\left\lbrack {x_{t}❘\mathcal{F}_{i - 1}} \right\rbrack} \\{E\left\lbrack {{x_{t}x_{t}^{T}}❘\mathcal{F}_{i - 1}} \right\rbrack}\end{pmatrix} = {\begin{pmatrix}{a_{t - 1}x_{t - 1}} \\{{a_{t - 1}{x_{t - 1}\left( {a_{t - 1}x_{t - 1}} \right)}^{T}} + {\sigma_{t - 1}\left( \sigma_{t - 1} \right)}^{T}}\end{pmatrix}.}} & (23)\end{matrix}$In this case, the block orthogonality condition vector is based on amultivariate stochastic system of difference equation and differencecalculus for (22) and (23), given by

$\begin{matrix}{{{H\left( {t_{i - 1},x_{t},x_{t - 1},a_{t - 1},\sigma_{t - 1}} \right)} = \begin{pmatrix}{x_{t} - {a_{t - 1}x_{t - 1}}} \\{{\Delta\;{V\left( x_{t} \right)}} - {{{LV}\left( {t,x_{t - 1}} \right)}\Delta\; t}}\end{pmatrix}},} & (24)\end{matrix}$where Δ and L are difference and L operators with respect toV=x_(t)x_(t) ^(T) for x∈

, and are defined by

$\begin{matrix}\left\{ {\begin{matrix}{{{\Delta\;{V\left( x_{t} \right)}} = {{V\left( x_{t} \right)} - {V\left( x_{t - 1} \right)}}},{{{for}\mspace{14mu} t} = 1},2,\ldots\mspace{14mu},t,\ldots\mspace{14mu},N} \\{{{LV}\left( {t,x_{t - 1}} \right)} = {{a_{t - 1}{x_{t - 1}\left( {\left( {2 + a_{t - 1}} \right)x_{t - 1}} \right)}^{T}} + {\sigma_{t - 1}\sigma_{t - 1}^{T}}}}\end{matrix},} \right. & (25)\end{matrix}$and differential of V with respect to multivariate difference system(22) parallel to continuous-time version (13) is as:ΔV(x _(t))=a _(t−1) x _(t−1)((2+a _(t−1))x _(t−1))^(T)+σ_(t−1),σ_(t−1)^(T)+2(1+a _(t−1) x _(t−1))(σ_(t−1) ,e _(t))^(T).  (26)

From the above, it is clear that the orthogonality condition parametervector in (24) is constructed with respect to multivariate stochasticsystem of difference equations and elementary difference calculus.

Remark 8.

From the transformation of system of stochastic differential equations(13) in Sub-section 3.1, the construction of Euler-type DiscretizationScheme for (12) and (13) in Sub-section 3.2, the Formation ofGeneralized Moment Equations from (15) in Sub-section 3.3, and the Basisfor Local Lagged Adaptive Discrete time Expectation Process inSub-section 3.4, the system is in the correct framework for mathematicalreasoning, logical, and interconnected/interactive within the context ofthe continuous-time dynamic system (12).

Further, a continuous-time state dynamic process described by systems ofstochastic differential equations (12) moves forward in time. Thetheoretical parameter estimation procedure in this section adapts to andincorporates the continuous-time changes in the state and parameters ofthe system and moves into a discrete time theoretical numerical schemesin (15) as a model validation of (12). It further successively moves inthe local moment equations within the context of local lagged adaptive,local discrete time statistic and computational processes in a natural,systematic, and coherent manner. On the other hand, the existing OCBGMMapproach is “single-shot” with a global approach, and it is highlydependent on the second component of the OCBGMM. That is, the use ofeither Euler-type discretization scheme or a discrete time econometricspecification regarding the stochastic differential equation. We referto OCBGMM as the single-shot or global approach with formation of asingle moment equation in a quadratic form.

Below, a result is stated that exhibits the existence of solution ofsystem of non linear algebraic equations. For the sake of reference, theImplicit Function Theorem is stated without proof.

Theorem 2 (Implicit Function Theorem).

Let F={F₁, F₂, . . . , F_(q)} be a vector-valued function defined on anopen set S∈

^(q+k) with values in

^(q). Suppose F∈C′ on S. Let (u₀; v₀) be a point in

for which F(u₀; v₀)=0 and for which the q×q determinant det[D_(j)F_(i)(u₀; v₀)]≠0. Then there exists a k-dimensional open set T₀containing v₀ and unique vector-valued function g, defined on T₀ andhaving values in

^(q), such that g∈C′ on T₀, g(v₀)=u₀, and F(g(v); v)=0 for every v∈T₀.

Illustration 1: Dynamic Model for Energy Commodity Price.

As one example, the stochastic dynamic model of energy commoditiesdescribed by the following nonlinear stochastic differential equation isconsidered:dy=ay(μ−y)dt+σ(t,y _(t))ydW(t),y _(t) ₀ =φ₀,  (27)where y_(t)(θ)=y(t+θ); θ∈[−τ,0], μ, a∈

, the initial process φ₀={y(t₀+θ)}_(θ∈[−τ,0]) is

—measurable and independent of {W(t),t∈[0,T]}, W(t) is a standard Wienerprocess defined in (12), σ:[0, T]×C→

is a Lipschitz continuous and bounded functional, and C is the Banachspace of continuous functions defined on [−τ,0] into

equipped with the supremum norm.

Transformation of Stochastic Differential Equation (27).

A Lyapunov function V(t,y)=ln(y) in (13) is picked for (27). UsingItô-differential formula,

$\begin{matrix}{{d\left( {\ln(y)} \right)} = {{\left\lbrack {{a\left( {\mu - y} \right)} - {\frac{1}{2}{\sigma^{2}\left( {t,y_{t}} \right)}}} \right\rbrack{dt}} + {{\sigma\left( {t,y_{t}} \right)}{{dW}.}}}} & (28)\end{matrix}$

The Euler-Type Discretization Schemes for (27) and (28).

By setting Δt_(i)=t_(i)−t_(i−1), Δy_(i)=y_(i)−y_(i−1), the combinedEuler discretized scheme for (27) and (28) is

$\begin{matrix}\left\{ {\begin{matrix}{\Delta\; y_{i}} & = & \begin{matrix}{{{{ay}_{i - 1}\left( {\mu - y_{i - 1}} \right)}\Delta\; t_{i}} +} \\{{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)y_{i - 1}\Delta\;{W\left( t_{i} \right)}},{y_{t_{0}} = \varphi_{0}},}\end{matrix} \\{\Delta\left( {\ln\left( y_{i} \right)} \right)} & = & \begin{matrix}\left\lbrack {{a\left( {\mu - y_{i - 1}} \right)} - {\frac{1}{2}{\sigma^{2}\left( {t_{i - 1},y_{t_{i - 1}}} \right)}}} \right\rbrack \\{{{\Delta\; t_{i}} + {{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)}\Delta\;{W\left( t_{i} \right)}}},{y_{t_{0}} = {\varphi_{0}.}}}\end{matrix}\end{matrix},} \right. & (29)\end{matrix}$where φ₀={y_(i)}_(i=−r) ⁰ is a given finite sequence of

measurable random variables, and it is independent of {ΔW(t_(i)}_(i=0)^(N).

Generalized Moment Equations.

Applying conditional expectation to (29) with respect to

$\begin{matrix}{{\mathcal{F}_{t_{i - 1}} \equiv \mathcal{F}_{i - 1}},\begin{matrix}{\left\lbrack {{\Delta\; y_{i}}❘\mathcal{F}_{i - 1}} \right\rbrack} & = & {{{ay}_{i - 1}\left( {\mu - y_{i - 1}} \right)}\Delta\; t} \\{\left\lbrack {{\Delta\left( {\ln\left( y_{i} \right)} \right)}❘\mathcal{F}_{i - 1}} \right\rbrack} & = & {\left\lbrack {{a\left( {\mu - y_{i - 1}} \right)} - {\frac{1}{2}{\sigma^{2}\left( {t_{i - 1},y_{t_{i - 1}}} \right)}}} \right\rbrack\Delta\;{t.}} \\ & = & {{\sigma^{2}\left( {t_{i - 1},y_{t_{i - 1}}} \right)}\Delta\;{t.}}\end{matrix}} & (30)\end{matrix}$

Basis for Lagged Adaptive Discrete Time Expectation Process.

From (30), (29) reduces to

$\begin{matrix}\left\{ {\begin{matrix}{\Delta\; y_{i}} & = & {{\left\lbrack {{\Delta\; y_{i}}❘\mathcal{F}_{i - 1}} \right\rbrack} + {{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)}y_{i - 1}\Delta\;{W\left( t_{i} \right)}}} \\{\Delta\left( {\ln\left( y_{i} \right)} \right)} & = & {{\left\lbrack {{\Delta\left( {\ln\left( y_{i} \right)} \right)}❘\mathcal{F}_{i - 1}} \right\rbrack} + {{\sigma\left( {t_{i - 1},y_{t_{i - 1}}} \right)}\Delta\;{W\left( t_{i} \right)}}}\end{matrix}.} \right. & (31)\end{matrix}$Equation (31) provides the basis for the development of the concept oflagged adaptive expectation process with respect to continuous timestochastic dynamic systems (27) and (28).

Remark 9. Orthogonality Condition Vector for (27) and (28).

Following Remark 7 and using (29), (30), and (31), the orthogonalitycondition vector with respect to continuous-time stochastic dynamicmodel (27) is represented by

$\begin{matrix}{{{H\left( {t_{i - 1},{y\left( t_{i} \right)},{y\left( t_{i - 1} \right)}} \right)} = \begin{pmatrix}{{\Delta\;{y\left( t_{i} \right)}} - {{{ay}\left( t_{i - 1} \right)}\left( {\mu - {y\left( t_{i - 1} \right)}} \right)\Delta\; t_{i}}} \\{{{\Delta ln}\left( {y\left( t_{i} \right)} \right)} - {L\;{\ln\left( {{y\left( t_{i - 1} \right)},y_{t_{i - 1}}} \right)}\Delta\; t_{i}}} \\{\left( {{{\Delta ln}\left( {y\left( t_{i} \right)} \right)} - {L\;{\ln\left( {{y\left( t_{i - 1} \right)},y_{t_{i - 1}}} \right)}\Delta\; t_{i}}} \right)^{2} - {{\sigma^{2}\left( {t_{i - 1},y_{t_{i - 1}}} \right)}\Delta\; t_{i}}}\end{pmatrix}},} & (32)\end{matrix}$wherein L ln

${\left( {{y\left( t_{i - 1} \right)},y_{t_{i - 1}}} \right)\Delta\; t_{i}} = {\left( {{a\left( {\mu - {y\left( t_{i - 1} \right)}} \right)} - {\frac{1}{2}{\sigma^{2}\left( {t_{i - 1},y_{t_{i - 1}}} \right)}}} \right)\Delta\;{t_{i}.}}$Unlike the orthogonality condition vector defined in the literature,this orthogonality condition vector is based on the discretizationscheme (29) associated with nonlinear continuous-time stochasticdifferential equations (27) and (28) and the Itô-Doob stochasticdifferential calculus.

Local Observation System of Algebraic Equations.

For k∈I₀(N), applying the lagged adaptive expectation process fromDefinitions 3-7, and using (8) and (31), a local observation/measurementprocess is formulated at t_(k) as one or more algebraic functions ofm_(k)-local restriction sequence of the overall finite sample sequence{y_(i)}_(i=−r) ^(N) to a subpartition P_(k) in Definition 2 as:

$\begin{matrix}\left\{ {{\begin{matrix}{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\;{\left\lbrack {{\Delta\; y_{i}}❘} \right\rbrack}}} & = & \begin{matrix}{a\left\lbrack {{\frac{\mu}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\; y_{i - 1}}} -} \right.} \\{{\left. {\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\; y_{i - 1}^{2}}} \right\rbrack\Delta\; t},}\end{matrix} \\{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\;{\left\lbrack {{\Delta\;\left( {\ln\left( y_{i} \right)} \right)}❘} \right\rbrack}}} & = & \begin{matrix}{{{a\left\lbrack {\mu - {\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\; y_{i - 1}}}} \right\rbrack}\Delta\; t} -} \\{\frac{1}{2m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\;\left\lbrack \left( {{\Delta\left( {\ln\left( y_{i} \right)} \right)} -} \right. \right.}} \\{\left. {\left. {\left\lbrack {{\Delta\left( {\ln\left( y_{i} \right)} \right)}❘} \right\rbrack} \right)^{2}❘} \right\rbrack,}\end{matrix}\end{matrix}{\hat{\sigma}}_{m_{k},k}^{2}} = \left\{ \begin{matrix}\begin{matrix}{\frac{1}{m_{k}\Delta\; t}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\;{\left( {{\Delta\left( {\ln\left( y_{i} \right)} \right)} -} \right.}}} \\\left. {\left. {\left\lbrack {{\Delta\left( {\ln\left( y_{i} \right)} \right)}❘} \right\rbrack} \right)^{2}❘} \right\rbrack\end{matrix} & {{if}\mspace{14mu} m_{k}\mspace{14mu}{is}\mspace{14mu}{small}} \\\begin{matrix}{\frac{1}{\left( {m_{k} - 1} \right)\Delta\; t}\sum\limits_{i = {k - m_{k}}}^{k - 1}} \\{\left( {{\Delta\left( {\ln\left( y_{i} \right)} \right)} -} \right.} \\\left. {\left. {\left\lbrack {{\Delta\left( {\ln\left( y_{i} \right)} \right)}❘} \right\rbrack} \right)^{2}❘} \right\rbrack\end{matrix} & {{if}\mspace{14mu} m_{k}\mspace{14mu}{is}\mspace{14mu}{{large}.}}\end{matrix} \right.} \right. & (33)\end{matrix}$

From the third equation in (33), it follows that the average volatilitysquare {circumflex over (σ)}_(m) _(k) _(,k) ² is given by

$\begin{matrix}{{{\hat{\sigma}}_{m_{k},k}^{2} = \frac{s_{m_{k},k}^{2}}{\Delta\; t}},} & (34)\end{matrix}$where s_(m) _(k) _(,k) ² is the local sample variance statistics forvolatility at t_(k) in the context of x(t_(i))=Δ(ln(y_(i))).

We define

$\begin{matrix}{{{F_{1}\left( {{\left\lbrack {{\Delta\; y_{i}}❘} \right\rbrack},{{\left\lbrack {{\Delta\left( {\ln\; y_{i}} \right)}❘} \right\rbrack};a},\mu} \right)} = {\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}\;{\left\lbrack {{\Delta\; y_{i}}❘} \right\rbrack}}{m_{k}} - {{a\left\lbrack {\frac{\mu{\sum\limits_{i = {k - m_{k}}}^{k - 1}\; y_{i - 1}}}{m_{k}} - \frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}\; y_{i - 1}^{2}}{m_{k}}} \right\rbrack}\Delta\; t}}}{{F_{2}\left( {{\left\lbrack {{\Delta\; y_{i}}❘} \right\rbrack},{{\left\lbrack {{\Delta\left( {\ln\; y_{i}} \right)}❘} \right\rbrack};a},\mu} \right)} = {{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\left\lbrack {{\Delta\left( {\ln\; y_{i}} \right)}❘} \right\rbrack}}} - {{a\left\lbrack {\mu - {\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}}}} \right\rbrack}\Delta\; t} + {\frac{s_{m_{k},k}^{2}}{2}.}}}} & (35)\end{matrix}$Then, we have

$\begin{matrix}\left\{ \begin{matrix}{F_{1}\left( {{\left\lbrack {{\Delta\; y_{i}}❘} \right\rbrack},{{\left\lbrack {{\Delta\left( {\ln\; y_{i}} \right)}❘} \right\rbrack};a},\mu} \right)} & {{= 0},} \\{F_{2}\left( {{\left\lbrack {{\Delta\; y_{i}}❘} \right\rbrack},{{\left\lbrack {{\Delta\left( {\ln\; y_{i}} \right)}❘} \right\rbrack};a},\mu} \right)} & {= 0.}\end{matrix} \right. & (36)\end{matrix}$

Let F={F₁, F₂}. The determinant of the Jacobian matrix of F is given by

$\begin{matrix}{{{{JF}\left( {a,\mu} \right)} = {{{- {\frac{a}{m_{k}}\left\lbrack {{\sum\limits_{i = {k - m_{k}}}^{k - 1}\; y_{i - 1}^{2}} - {\frac{1}{m_{k}}\left( {\sum\limits_{i = {k - m_{k}}}^{k - 1}\; y_{i - 1}} \right)^{2}}} \right\rbrack}}\left( {\Delta\; t} \right)^{2}} = {{{- a}\;{{var}\left( {y\left( t_{i - 1} \right)}_{i = {k - m_{k}}}^{k - 1} \right)}\left( {\Delta\; t} \right)^{2}} \neq 0}}},} & (37)\end{matrix}$provided that a≠0 or the sequence {x(t_(i−1))}_(i=−r+1) ^(N) is neitherzero nor a constant. This fulfils the hypothesis of Theorem 2.

Thus, by the application of Theorem 2 (Implicit Function Theorem), weconclude that for every non-constant m_(k)-local sequence{x(t_(i))}_(i=k−m) _(k) ^(k−1), there exists a unique solution of systemof algebraic equations (36), â_(m) _(k) _(,k) and {circumflex over(μ)}_(m) _(k) _(,k) as a point estimates of a and μ, respectively.

We also note that the estimated values of a and μ change at each timet_(k). For instance, at time t₀=0 and the given

measurable discrete time process y_(−r+1), y_(−r+2), . . . , y⁻¹, (33)reduces to

$\begin{matrix}\left\{ \begin{matrix}{\frac{1}{m_{0}}{\sum\limits_{i = {- m_{0}}}^{0}{\Delta\; y_{i}}}} & = & {{{a\left\lbrack {{\frac{\mu}{m_{0}}{\sum\limits_{i = {- m_{0}}}^{0}y_{i - 1}}} - {\frac{1}{m_{0}}{\sum\limits_{i = {- m_{0}}}^{0}y_{i - 1}^{2}}}} \right\rbrack}\Delta\; t},} \\{\frac{1}{m_{0}}{\sum\limits_{i = {- m_{0}}}^{0}{\Delta\;\left( {\ln\; y_{i}} \right)}}} & = & {{{{a\left\lbrack {\mu - {\frac{1}{m_{0}}{\sum\limits_{i = {- m_{0}}}^{0}y_{i - 1}}}} \right\rbrack}\Delta\; t} - \frac{s_{m_{0},0}^{2}}{2}},} \\{\hat{\sigma}}_{m_{0},0}^{2} & = & {\frac{s_{m_{0},0}^{2}}{\Delta\; t}.}\end{matrix} \right. & (38)\end{matrix}$

The initial solution of algebraic equations (38) at time t₀ is given by

$\begin{matrix}\left\{ {\begin{matrix}{\hat{a}}_{m_{0},0} & = & \frac{\begin{matrix}{{\left( {{\frac{1}{m_{0}}{\sum\limits_{i = {- m_{0}}}^{0}\;{\Delta\left( {\ln\; y_{i}} \right)}}} + \frac{s_{m_{0},0}^{2}}{2}} \right)\left( {\frac{1}{m_{0}}{\sum\limits_{i = {- m_{0}}}^{0}y_{i - 1}}} \right)} -} \\{\frac{1}{m_{0}}{\sum\limits_{i = {- m_{0}}}^{0}{\Delta\; y_{i}}}}\end{matrix}}{{\frac{1}{m_{0}}\left\lbrack {{\sum\limits_{i = {- m_{0}}}^{0}y_{i - 1}^{2}} - {\frac{1}{m_{0}}\left( {\sum\limits_{i = {- m_{0}}}^{0}y_{i - 1}} \right)^{2}}} \right\rbrack}\Delta\; t} \\{\hat{\mu}}_{m_{0},0} & = & \frac{{\frac{1}{m_{0}\Delta\; t}{\sum\limits_{i = {- m_{0}}}^{0}{\Delta\left( {\ln\; y_{i}} \right)}}} + \frac{s_{m_{0},0}^{2}}{2\Delta\; t} + {\frac{{\hat{a}}_{m_{0},0}}{m_{0}}\left( {\sum\limits_{i = {- m_{0}}}^{0}y_{i - 1}} \right)}}{{\hat{a}}_{m_{0},0}} \\{\hat{\sigma}}_{m_{0},0}^{2} & = & \frac{s_{m_{0},0}^{2}}{\Delta\; t}\end{matrix}.} \right. & (39)\end{matrix}$

At time t₁=1 and the given

₀ measurable discrete time process y_(−r), y_(−r+1), . . . , y⁻¹, y₀,(33) reduces to

$\begin{matrix}\left\{ \begin{matrix}{\frac{1}{m_{1}}{\sum\limits_{i = {1 - m_{1}}}^{0}\;{\Delta\; y_{i}}}} & = & {{{a\left\lbrack {{\frac{\mu}{m_{1}}{\sum\limits_{i = {1 - m_{1}}}^{0}y_{i - 1}}} - {\frac{1}{m_{1}}{\sum\limits_{i = {1 - m_{1}}}^{0}y_{i - 1}^{2}}}} \right\rbrack}\Delta\; t},} \\{\frac{1}{m_{1}}{\sum\limits_{i = {1 - m_{1}}}^{0}{\Delta\left( {\ln\; y_{i}} \right)}}} & = & {{{{a\left\lbrack {\mu - {\frac{1}{m_{1}}{\sum\limits_{i = {1 - m_{1}}}^{0}y_{i - 1}}}} \right\rbrack}\Delta\; t} - \frac{s_{m_{1},1}^{2}}{2}},} \\{\hat{\sigma}}_{m_{1},1}^{2} & = & {\frac{s_{m_{1},1}^{2}}{\Delta\; t}.}\end{matrix} \right. & (40)\end{matrix}$

The solution of algebraic equations (40) is given by

$\begin{matrix}\left\{ {\begin{matrix}{\hat{a}}_{m_{1},1} & = & \frac{\begin{matrix}{{\left( {{\frac{1}{m_{1}}{\sum\limits_{i = {1 - m_{1}}}^{0}\;{\Delta\left( {\ln\; y_{i}} \right)}}} + \frac{s_{m_{1},1}^{2}}{2}} \right)\left( {\frac{1}{m_{1}}{\sum\limits_{i = {1 - m_{1}}}^{0}y_{i - 1}}} \right)} -} \\{\frac{1}{m_{1}}{\sum\limits_{i = {1 - m_{1}}}^{0}{\Delta\; y_{i}}}}\end{matrix}}{{\frac{1}{m_{1}}\left\lbrack {{\sum\limits_{i = {1 - m_{1}}}^{0}y_{i - 1}^{2}} - {\frac{1}{m_{1}}\left( {\sum\limits_{i = {1 - m_{1}}}^{0}y_{i - 1}} \right)^{2}}} \right\rbrack}\Delta\; t} \\{\hat{\mu}}_{m_{1},1} & = & \frac{{\frac{1}{m_{1}\Delta\; t}{\sum\limits_{i = {1 - m_{1}}}^{0}{\Delta\left( {\ln\; y_{i}} \right)}}} + \frac{s_{m_{1},1}^{2}}{2\Delta\; t} + {\frac{{\hat{a}}_{m_{1},1}}{m_{1}}\left( {\sum\limits_{i = {1 - m_{1}}}^{0}y_{i - 1}} \right)}}{{\hat{a}}_{m_{1},1}} \\{\hat{\sigma}}_{m_{1},1}^{2} & = & \frac{s_{m_{1},1}^{2}}{\Delta\; t}\end{matrix}.} \right. & (41)\end{matrix}$

Likewise, for k=2, we have

$\begin{matrix}\left\{ \begin{matrix}{\hat{a}}_{m_{2},2} & = & \frac{\begin{matrix}{{\left( {{\frac{1}{m_{2}}{\sum\limits_{i = {2 - m_{2}}}^{1}\;{\Delta\left( {\ln\; y_{i}} \right)}}} + \frac{s_{m_{k},k}^{2}}{2}} \right)\left( {\frac{1}{m_{2}}{\sum\limits_{i = {2 - m_{2}}}^{1}y_{i - 1}}} \right)} -} \\{\frac{1}{m_{2}}{\sum\limits_{i = {2 - m_{2}}}^{1}{\Delta\; y_{i}}}}\end{matrix}}{{\frac{1}{m_{2}}\left\lbrack {{\sum\limits_{i = {2 - m_{2}}}^{1}y_{i - 1}^{2}} - {\frac{1}{m_{2}}\left( {\sum\limits_{i = {2 - m_{2}}}^{1}y_{i - 1}} \right)^{2}}} \right\rbrack}\Delta\; t} \\{\hat{\mu}}_{m_{2},2} & = & {\frac{{\frac{1}{m_{2}\Delta\; t}{\sum\limits_{i = {2 - m_{2}}}^{1}{\Delta\left( {\ln\; y_{i}} \right)}}} + \frac{s_{m_{2},2}^{2}}{2\Delta\; t} + {\frac{{\hat{a}}_{m_{2},2}}{m_{2}}\left( {\sum\limits_{i = {2 - m_{2}}}^{1}y_{i - 1}} \right)}}{{\hat{a}}_{m_{2},2}},} \\{\hat{\sigma}}_{m_{2},2}^{2} & = & {\frac{s_{m_{2},2}^{2}}{\Delta\; t}.}\end{matrix} \right. & (42)\end{matrix}$

Hence, from (33) and applying the principle of mathematical induction,we have

$\begin{matrix}\left\{ \begin{matrix}{\hat{a}}_{m_{k},k} & = & \frac{\begin{matrix}{{\left( {{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\;{\Delta\left( {\ln\; y_{i}} \right)}}} + \frac{s_{m_{k},k}^{2}}{2}} \right)\left( {\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}}} \right)} -} \\{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\Delta\; y_{i}}}}\end{matrix}}{{\frac{1}{m_{k}}\left\lbrack {{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{2}} - {\frac{1}{m_{k}}\left( {\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}} \right)^{2}}} \right\rbrack}\Delta\; t} \\{\hat{\mu}}_{m_{k},k} & = & {\frac{{\frac{1}{m_{k}\Delta\; t}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\Delta\left( {\ln\; y_{i}} \right)}}} + \frac{s_{m_{k},k}^{2}}{2\Delta\; t} + {\frac{{\hat{a}}_{m_{k},k}}{m_{k}}\left( {\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}} \right)}}{{\hat{a}}_{m_{k},k}},} \\{\hat{\sigma}}_{m_{k},k}^{2} & = & {\frac{s_{m_{k},k}^{2}}{\Delta\; t}.}\end{matrix} \right. & (43)\end{matrix}$

Remark 10.

We note that without loss in generality, the discrete time data set{y_(−r+i): i∈I₁(r−1)} is assumed to be close to the true values of thesolution process of the continuous-time dynamic process. This assumptionis feasible in view of the uniqueness and continuous dependence ofsolution process of stochastic functional or ordinary differentialequation with respect to the initial data.

Remark 11.

If the sample {y_(i)}_(i=k−m) _(k) ⁻¹ ^(k−1) is a constant sequence,then it follows from (43) and the fact that Δ(ln y_(i))=0 and s_(m) _(k)_(,k) ²=0, that

$\left. {\hat{\mu}}_{m_{k},k}\rightarrow{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{y_{i - 1}.}}} \right.$Hence, it follows from (33) that â_(m) _(k) _(,k)=0.

Remark 12.

The estimated parameters a, μ, and σ² depend upon the time at which datapoint is drawn. This is expected because of the nonlinearity of thedynamic model together with environmental stochastic perturbationsgenerate non stationary solution process. Using this locally estimatedparameters of the continuous-time dynamic system, we can find theaverage of these local parameters over the entire size of data set asfollows:

$\begin{matrix}\left\{ \begin{matrix}\overset{\_}{a} & = & {{\frac{1}{N}{\sum\limits_{i = 0}^{N}\; a_{{\hat{m}}_{i},i}}},} \\\overset{\_}{\mu} & = & {\frac{1}{N}{\sum\limits_{i = 0}^{N}\;\mu_{{\hat{m}}_{i},i}}} \\\overset{\_}{\sigma^{2}} & = & {\frac{1}{N}{\sum\limits_{i = 0}^{N}\;{\sigma_{{\hat{m}}_{i},i}^{2}.}}}\end{matrix} \right. & (44)\end{matrix}$

Here, ā, μ, and σ² are referred to as aggregated parameter estimates ofa, μ, and σ² over the given entire finite interval of time,respectively.

Remark 13.

The DTIDMLSMVSP and its transformation of data are utilized in (33),(34), (35), (43), and (44) for updating statistic coefficients ofequations in (30). This accelerates the computation process.Furthermore, the DTIDMLSMVSP plays a significant role in the localdiscretization and model validation errors.

Illustration 2: Dynamic Model for U.S. Treasury Bill Interest Rate andthe USD-EUR Exchange Rate.

As noted above, at reference numeral 204 in FIG. 2, the process 200includes developing a stochastic model of a continuous time dynamicprocess. As another example of this, the scheme presented above can beapplied for estimating parameters of a continuous-time model for U.S.Treasury Bill Interest Rate and USD-EUR Exchange Rate processes. Byemploying dynamic modeling process, a continuous time dynamic model ofinterest rate process under random environmental perturbations can bedescribed bydy=(βy+μy ^(δ))dt+σy ^(γ) dW(t),y(t ₀)=y ₀,  (45)where β, μ, δ, σ, γ∈

; y(t, t₀, y₀) is adapted, non-anticipating solution process withrespect to

, the initial process y₀ is

measurable and independent of {W(t), t∈[t₀, T] }, and W(t) is a standardWiener process defined on a filtered probability space (Ω,

).

Transformation of Stochastic Differential Equation (45).

As part of the conceptual aspects of generating the DTIDMLSMVSP, atreference numeral 206 in FIG. 2, the process 200 can includetransforming the stochastic model of the continuous time dynamic processinto a stochastic model of a discrete time dynamic process utilizing adiscretization scheme. As another example of this, for (45), theLyapunov functions V₁(t,y)=½y² and V₂(t,y)=⅓y³ as in (13) areconsidered. The Itô-differentials of V_(i), for i=1, 2, are given by

$\begin{matrix}\left\{ {\begin{matrix}{{dV}_{1} = {{\left\lbrack {{y\left( {{\beta\; y} + {\mu\; y^{\delta}}} \right)} + {\frac{1}{2}\sigma^{2}y^{2\gamma}}} \right\rbrack{dt}} + {\sigma\; y^{\gamma + 1}{dW}}}} \\{{dV}_{2} = {{\left\lbrack {{y^{2}\left( {{\beta\; y} + {\mu\; y^{\delta}}} \right)} + {\sigma^{2}y^{{2\gamma} + 1}}} \right\rbrack{dt}} + {\sigma\; y^{\gamma + 2}{dW}}}}\end{matrix}.} \right. & (46)\end{matrix}$

The Euler-Type Numerical Schemes for (45) and (46)).

Following the approach in Section 3.5, the Euler discretized scheme(Delta=1) for (45) is defined by

$\begin{matrix}\left\{ {\begin{matrix}{\Delta\; y_{i}} & = & {\left( {{\beta\; y_{i - 1}} + {\mu\; y_{i - 1}^{\delta}}} \right) + {\sigma\; y_{i - 1}^{\gamma}\Delta\;{W\left( t_{i} \right)}}} \\{\frac{1}{2}\Delta\;\left( y_{i}^{2} \right)} & = & {{y_{i - 1}\left( {{\beta\; y_{i - 1}} + {\mu\; y_{i - 1}^{\delta}}} \right)} + {\frac{1}{2}\sigma^{2}y_{i - 1}^{2\;\gamma}} + {\sigma\; y_{i - 1}^{\gamma + 1}\Delta\; W_{i}}} \\{\frac{1}{3}{\Delta\left( y_{i}^{3} \right)}} & = & {{y_{i - 1}^{2}\left( {{\beta\; y_{i - 1}} + {\mu\; y_{i - 1}^{\delta}}} \right)} + {\sigma^{2}y_{i - 1}^{{2\gamma} + 1}} + {\sigma\; y_{i - 1}^{\gamma + 2}\Delta\; W_{i}}}\end{matrix}.\quad} \right. & (47)\end{matrix}$

Generalized Moment Equations.

As another part of the conceptual aspects of generating the DTIDMLSMVSP,at reference numeral 206 in FIG. 2, the process 200 can also includedeveloping a system of generalized method of moments equations from thestochastic model of the discrete time dynamic process. As anotherexample of this, applying conditional expectation to (47) with respectto

_(i−1),

$\begin{matrix}{\begin{matrix}{\left\lbrack {\Delta\; y_{i}} \middle| \mathcal{F}_{i - 1} \right\rbrack} & = & {{\beta\; y_{i - 1}} + {\mu\; y_{i - 1}^{\delta}}} \\{\frac{1}{2}{\left\lbrack {\Delta\;\left( y_{i}^{2} \right)} \middle| \mathcal{F}_{i - 1} \right\rbrack}} & = & {{\beta\; y_{i - 1}^{2}} + {\mu\; y_{i - 1}^{\delta + 1}} + {\frac{1}{2}\sigma^{2}y_{i - 1}^{2\gamma}}} \\{\frac{1}{3}{\left\lbrack {\Delta\;\left( y_{i}^{3} \right)} \middle| \mathcal{F}_{i - 1} \right\rbrack}} & = & {{\beta\; y_{i - 1}^{3}} + {\mu\; y_{i - 1}^{\delta + 2}} + {\frac{1}{2}\sigma^{2}y_{i - 1}^{{2\gamma} + 1}}} \\{\left\lbrack \left( {{\Delta\; y_{i}} - {\left\lbrack {\Delta y}_{i} \middle| \mathcal{F}_{i - 1} \right\rbrack}} \right)^{2} \middle| \mathcal{F}_{i - 1} \right\rbrack} & = & {{\sigma^{2}y_{i - 1}^{2\gamma}},} \\{\frac{1}{4}{\left\lbrack \left( {\Delta\;\left( y_{i}^{2} \right){\left\lbrack {\Delta\left( y_{i}^{2} \right)} \right\rbrack}} \right)^{2} \middle| \mathcal{F}_{i - 1} \right\rbrack}} & = & {\sigma^{2}y_{i - 1}^{{2\gamma} + 2}}\end{matrix}.} & (48)\end{matrix}$

Basis for Lagged Adaptive Discrete Time Expectation Process.

From (48), (47) reduces to

$\begin{matrix}\left\{ {\begin{matrix}{\Delta\; y_{i}} & = & {{\left\lbrack {\Delta\; y_{i}} \middle| \mathcal{F}_{i - 1} \right\rbrack} + {\sigma\; y_{i - 1}^{\gamma}\Delta\;{W\left( t_{i} \right)}}} \\{\frac{1}{2}\Delta\;\left( y_{i}^{2} \right)} & = & {{\frac{1}{2}{\left\lbrack {\Delta\;\left( y_{i}^{2} \right)} \middle| \mathcal{F}_{i - 1} \right\rbrack}} + {\sigma\; y_{i - 1}^{\gamma + 1}\Delta\; W_{i}}} \\{\frac{1}{3}{\Delta\left( y_{i}^{3} \right)}} & = & {{\frac{1}{3}{\left\lbrack {\Delta\;\left( y_{i}^{3} \right)} \middle| \mathcal{F}_{i - 1} \right\rbrack}} + {{\sigma y}_{i - 1}^{\gamma + 2}\Delta\; W_{i}}}\end{matrix}{\quad.}} \right. & (49)\end{matrix}$

Remark 14. (Orthogonality Condition Vector for (45) and (46)).

Again, imitating Remarks 7, 8 and 9 in the context of (45), (46), (47),(48), and (49), the orthogonality condition vector with respect to thecontinuous-time stochastic dynamic model (45) is

$\begin{matrix}{{{H\left( {t_{i - 1},{y\left( t_{i} \right)},{y\left( t_{i - 1} \right)}} \right)} = \begin{pmatrix}{{\Delta\;{y\left( t_{i} \right)}} - {\left( {{\beta\;{y\left( t_{i - 1} \right)}} + {\mu\;{y^{\delta}\left( t_{i - 1} \right)}}} \right)\Delta\; t_{i}}} \\{{\frac{1}{2}\Delta\;\left( {y^{2}\left( t_{i} \right)} \right)} - {{L\left( {y^{2}\left( t_{i - 1} \right)} \right)}\Delta\; t_{i}}} \\{{\frac{1}{3}\Delta\;\left( {y^{3}\left( t_{i} \right)} \right)} - {{L\left( {y^{3}\left( t_{i - 1} \right)} \right)}\Delta\; t_{i}}} \\{\left( {{y\left( t_{i} \right)} - {\left( {{\beta\;{y\left( t_{i - 1} \right)}} + {\mu\;{y^{\delta}\left( t_{i - 1} \right)}}} \right)\Delta\; t_{i}}} \right)^{2} - {\sigma^{2}{y^{2\;\gamma}\left( t_{i - 1} \right)}\Delta\; t_{i}}} \\{\left( {{\frac{1}{2}{\Delta\left( {y^{2}\left( t_{i} \right)} \right)}} - {{L\left( {y^{2}\left( t_{i - 1} \right)} \right)}\Delta\; t_{i}}} \right)^{2} - {\sigma^{2}{y^{{2\gamma} + 2}\left( t_{i - 1} \right)}\Delta\; t_{i}}}\end{pmatrix}},} & (50)\end{matrix}$where

${{L\left( {y^{2}\left( t_{i - 1} \right)} \right)}\Delta\; t_{i}} = {\left( {{{y\left( t_{i - 1} \right)}\left( {{\beta\;{y\left( t_{i - 1} \right)}} + \;{\mu\;{y^{\delta}\left( t_{i - 1} \right)}}} \right)} + {\frac{1}{2}\sigma^{2}{y^{2\;\gamma}\left( t_{i - 1} \right)}}} \right)\Delta\; t_{i}}$and

L(y³(t_(i − 1)))Δ t_(i) = (y²(t_(i − 1))(β y(t_(i − 1))+ μ y^(δ)(t_(i − 1))) + σ²y^(2 γ + 1)(t_(i − 1)))Δ t_(i).Further, unlike other orthogonality condition vectors, thisorthogonality condition vector is based on the discretization scheme(47) associated with nonlinear continuous-time stochastic differentialequations (45) and (46).

Local Observation System of Algebraic Equations.

Following the argument used in (33), for k∈I₀(N), applying the laggedadaptive expectation process from Definitions 3-7, and using (8) and(48), we formulate a local observation/measurement process at t_(k) as aalgebraic functions of m_(k)-local functions of restriction of theoverall finite sample sequence {y_(i)}_(i=−r) ^(N) to subpartition P inDefinition 2, as

$\begin{matrix}{\begin{matrix}{{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\left\lbrack \left. {\Delta\; y_{i}} \right| \right\rbrack}}} = {{\beta\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}}{m_{k}}} + {\mu\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta}}{m_{k}}}}} \\{\begin{matrix}{\frac{1}{2m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\left\lbrack {{\left\lbrack \left. {\Delta\left( \; y_{i}^{2} \right)} \right| \right\rbrack} -} \right.}} \\\left. {\left\lbrack \left. \left( {\Delta\; y_{i}{\left\lbrack \left. {\Delta\; y_{i}} \right| \right\rbrack}} \right)^{2} \right| \right\rbrack} \right\rbrack\end{matrix} = {{\beta\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{2}}{m_{k}}} + {\mu\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta + 1}}{m_{k}}}}} \\{{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\left\lbrack {{\frac{1}{3}{\left\lbrack \left. {\Delta\left( \; y_{i}^{3} \right)} \right| \right\rbrack}} - {\sigma^{2}y_{i - 1}^{{2\gamma} + 1}}} \right\rbrack}} = {{\beta\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{3}}{m_{k}}} + {\mu\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta + 2}}{m_{k}}}}}\end{matrix}\begin{matrix}{{{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\left\lbrack \left. \left( {\Delta\; y_{i}{\left\lbrack \left. {\Delta\; y_{i}} \right| \right\rbrack}} \right)^{2} \right| \right\rbrack}}} = {\sigma^{2}\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{2\gamma}}{m_{k}}}},} \\{{\frac{1}{4m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\left\lbrack \left. \left( {{\Delta\left( \; y_{i}^{2} \right)} - {\left\lbrack {\Delta\left( y_{i}^{2} \right)} \right\rbrack}} \right)^{2} \right| \right\rbrack}}} = {\sigma^{2}{\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{{2\gamma} + 2}}{m_{k}}.}}}\end{matrix}} & (51)\end{matrix}$

Following the approach discussed in Section 5, the solution of σ_(m)_(k) _(,k) is given by

$\begin{matrix}{{\sigma_{m_{k},k} = \left\lbrack \frac{s_{m_{k},k}^{2}}{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{2\;\gamma_{m_{k},k}}}} \right\rbrack^{1/2}},} & (52)\end{matrix}$and γ_(m) _(k) _(,k) satisfies the following nonlinear algebraicequation

$\begin{matrix}{{{{s_{m_{k},k}^{2}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{{2\;\gamma_{m_{k},k}} + 2}}} - {\frac{1}{4}s_{m_{k},k}^{2}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{2\;\gamma_{m_{k},k}}}}} = 0},} & (53)\end{matrix}$where s_(m) _(k) _(,k) ², and s_(m) _(k) _(,k) ² denotes the localmoving variance of Δy_(i) and Δ(y_(i) ²) respectively.

To solve for the parameters β and μ, and δ, we define the conditionalmoment functions

F_(j) ≡ F_(j)([Δ y_(i)|ℱ_(i − 1)], [Δ (y_(i))²|ℱ_(i − 1)], [Δ (y_(i))³|ℱ_(i − 1)]), j = 1, 2, 3as

$\begin{matrix}\begin{matrix}{F_{1} = {{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\left\lbrack {\Delta\; y_{i}} \middle| \mathcal{F}_{i - 1} \right\rbrack}}} - {\beta\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}}{m_{k}}} - {\mu\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta}}{m_{k}}}}} \\{F_{2} = \begin{matrix}{\frac{1}{2m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\left\lbrack {{\left\lbrack {\Delta\;\left( y_{i}^{2} \right)} \middle| \mathcal{F}_{i - 1} \right\rbrack},\left\lbrack \left( {{\Delta\; y_{i}} -} \right. \right.} \right.}} \\{\left. \left. \left\lbrack \left( {\Delta\; y_{i}} \middle| \mathcal{F}_{i - 1} \right\rbrack \right)^{2} \middle| \mathcal{F}_{i - 1} \right\rbrack \right\rbrack - {\beta\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{2}}{m_{k}}} - {\mu\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta + 1}}{m_{k}}}}\end{matrix}} \\{F_{3} = \begin{matrix}{{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}\left\lbrack {{\frac{1}{3}{\left\lbrack {\Delta\;\left( y_{i}^{3} \right)} \middle| \mathcal{F}_{i - 1} \right\rbrack}} - {\sigma^{2}y_{i - 1}^{{2\;\gamma} + 1}}} \right\rbrack}} -} \\{{\beta\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{3}}{m_{k}}} - {\mu{\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta + 2}}{m_{k}}.}}}\end{matrix}}\end{matrix} & (54)\end{matrix}$

Using (51), we have

$\begin{matrix}\left\{ {\begin{matrix}{F_{1} = 0} \\{F_{2} = 0} \\{F_{3} = 0}\end{matrix}.} \right. & (55)\end{matrix}$

Let F={F₁, F₂, F₃}. The determinant of the Jacobian matrix of F is givenby

$\begin{matrix}{{{{JF}\left( {\beta,\mu,\delta} \right)} = {{{- \frac{1}{m_{k}^{3}}}{\det\begin{pmatrix}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}} & {\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta}} & {\sum\limits_{i = {k - m_{k}}}^{k - 1}{\left( {\ln\; y_{i - 1}} \right)y_{i - 1}^{\delta}}} \\{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{2}} & {\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta + 1}} & {\sum\limits_{i = {k - m_{k}}}^{k - 1}{\left( {\ln\; y_{i - 1}} \right)y_{i - 1}^{\delta + 1}}} \\{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{3}} & {\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta + 2}} & {\sum\limits_{i = {k - m_{k}}}^{k - 1}{\left( {\ln\; y_{i - 1}} \right)y_{i - 1}^{\delta + 2}}}\end{pmatrix}}} \neq 0}},} & (56)\end{matrix}$provided δ≠1 and the sequence {y(t_(i−1))}_(i=k−m) _(k) ^(k−1) isneither zero nor a constant sequence. Thus, by the application ofTheorem 2 (Implicit Function Theorem), we conclude that for everynon-constant m_(k)-local sequence {y(t_(i))}_(i=k−m) _(k) ^(k−1), δ≠1,there exist a solution of system of algebraic equations (55) {circumflexover (β)}_(m) _(k) _(,k), {circumflex over (μ)}_(m) _(k) _(,k−1),{circumflex over (δ)}_(m) _(k) _(,k) as a point estimates of β and μ,and δ respectively.

The solution of system of algebraic equations (55) is given by

$\begin{matrix}\left\{ {\begin{matrix}{{\hat{\mu}}_{m_{k},k} = \frac{\begin{matrix}{{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\Delta\; y_{i}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{2}}}}} -} \\{{\frac{1}{2}\left\lbrack {{\frac{1}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\Delta\left( y_{i}^{2} \right)}}} - s_{m_{k},k}^{2}} \right\rbrack}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}}}\end{matrix}}{\frac{1}{m_{k}}\left\lbrack {{\sum\limits_{i = {k - m_{k}}}^{k - 1}{y_{i - 1}^{\delta_{m_{k},k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{2}}}} - {\sum\limits_{i = {k - m_{k}}}^{k - 1}{y_{i - 1}^{1 + \delta_{m_{k},k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}}}}} \right\rbrack}} \\{{\hat{\beta}}_{m_{k},k} = \frac{{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\Delta\; y_{i}}} - {{\hat{\mu}}_{m_{k},k}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta_{m_{k},k}}}}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}}}\end{matrix},} \right. & (57)\end{matrix}$where δ_(m) _(k) _(,k) satisfies the third equation in (51) described by

$\begin{matrix}{{{\frac{1}{3m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}{\Delta\;\left( y_{i}^{3} \right)}}} - {\frac{\sigma_{m_{k},k}^{2}}{m_{k}}{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{{2\;\gamma_{m_{k},k}} + 1}}} - {\beta\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{3}}{m_{k}}} - {\mu\frac{\sum\limits_{i = {k - m_{k}}}^{k - 1}y_{i - 1}^{\delta + 2}}{m_{k}}}} = 0} & (58)\end{matrix}$

The parameters of continuous-time dynamic process described by (45) aretime-varying functions. This justifies the modifications/correctnessneeded for the development of continuous-time models of dynamicprocesses.

Remark 15.

The illustrations presented above exhibit the important featuresdescribed in Remark 8 of the theoretical parameter estimation procedure.The illustrations further clearly differentiate the Itô-Doobdifferential formula based formation of orthogonality condition vectorsin Remarks 9 and 14 and the algebraic manipulation and discretizedscheme using the econometric specification based orthogonality conditionvectors.

Remark 16. The DTIDMLSMVSP and its transformation of data are utilizedin (51), (52), (53), (57), and (58) for updating statistic coefficientof equations in (45). Again, this accelerates the computation process.Furthermore, the DTIDMLSMVSP plays a significant role in the localdiscretization and model validation errors.

4. Computational Algorithm

In this section, the computational, data organizational, and simulationschemes are outlined. The ideas of iterative data process and datasimulation process time schedules in relation with the real time dataobservation/collection schedule are also introduced. For thecomputational estimation of continuous time stochastic dynamic systemstate and parameters, it is important to determine an admissible set oflocal conditional sample average and sample variance, in particular, thesize of local conditional sample in the context of a partition of timeinterval [−τ, T]. Further, the discrete time dynamic model ofconditional sample mean and sample variance statistic processes inSection 2 and the theoretical parameter estimation scheme in Section 3coupled with the lagged adaptive expectation process motivate to outlinea computational scheme in a systematic and coherent manner. A briefconceptual computational scheme and simulation process summary isdescribed below:

4.1. Coordination of Data Observation, Iterative Process, and SimulationSchedules

Without loss of generality, we assume that the real dataobservation/collection partition schedule P is defined in (2). Now, wepresent definitions of iterative process and simulation time schedules.

Definition 8.

The iterative process time schedule in relation with the real datacollection schedule is defined byIP={F ^(−r) t _(i): for t _(i) ∈P},  (59)where F^(−r)t_(i)=t_(i−r), and F^(−r) is a forward shift operator.

The simulation time is based on the order p of the time series model ofm_(k)-local conditional sample mean and variance processes in Lemma 1 inSection 2.

Definition 9.

The simulation process time schedule in relation with the real dataobservation schedule is defined by

$\begin{matrix}{{SP} = \left\{ {\begin{matrix}{\left\{ {F^{\; r}{t_{i}:\;{{{for}\mspace{14mu} t_{i}} \in P}}} \right\},\;{{{if}\mspace{11mu} p}\; \leq r}} \\{\left\{ {F^{p}{t_{i}:\;{{{for}\mspace{14mu} t_{i}} \in P}}} \right\},\;{{{if}\mspace{11mu} p} > r}}\end{matrix}.} \right.} & (60)\end{matrix}$

Remark 17.

The initial times of iterative and simulation processes are equal to thereal data times t_(r) and t_(p), respectively. Further, iterative andsimulation process times in (59) and (60), respectively, justify Remark10. In short, t_(i) is the scheduled time clock for the collection ofthe i th observation of the state of the system under investigation. Theiterative process and simulation process times are t_(i+r) and t_(i+p),respectively.

4.2. Computational Parameter Estimation Scheme

For the conceptual computational dynamic system parameter estimation, afew concepts are introduced below, including local admissiblesample/data observation size, m_(k)-local admissible conditional finitesequence at t_(k)∈SP, and local finite sequence of parameter estimatesat t_(k).

Referring back to the drawings, as part of the computational aspects ofgenerating the DTIDMLSMVSP at reference numeral 206 (FIG. 2), in FIG. 3the process includes selecting at least one partition P in the timeinterval [−τ,0] of the discrete time data set [−τ,T] as past stateinformation of a continuous time dynamic process at reference numeral302. As described herein, multiple partitions P in the time interval[−τ,0] can be selected in the iterative, nested process.

Definition 10.

For each k∈I₀(N), we define local admissible sample/data observationsize m_(k) at t_(k) as m_(k)∈OS_(k), where

$\begin{matrix}{{OS}_{k} = \left\{ {\begin{matrix}{{I_{2}\left( {r + k - 1} \right)},\;{{{if}\mspace{11mu} p}\; \leq r},} \\{{{I_{2}\left( {p + k - 1} \right)},\;{{{if}\mspace{11mu} p}\; > r},}\;}\end{matrix}.} \right.} & (61)\end{matrix}$Further, OS_(k) is referred as the local admissible set of laggedsample/data observation size at t_(k). In other words, at referencenumeral 304 in FIG. 3, at each time point in the partition P, theprocess includes selecting an m_(k)-point sub-partition P_(k) of thepartition P, the m_(k)-point sub-partition having a local admissiblelagged sample observation size OS_(k) based on p, r, and a sub-partitiontime observation index size k.

Definition 11.

For each admissible m_(k)∈OS_(k) in Definition 10, an m_(k)-localadmissible lagged-adapted finite restriction sequence of conditionalsample/data observation at t_(k) to subpartition P_(k) of P inDefinition 3 is defined by {

[y_(i)|

]}_(i=k−m) _(k) ^(k−1). Further, an m_(k)-class of admissiblelagged-adapted finite sequences of conditional sample/data observationof size m_(k) at t_(k) is defined by

={{

[y _(i)|

]}_(i=k−m) _(k) ^(k−1) : m _(k) ∈OS _(k)}={{

[y _(i)|

]}_(i=k−m) _(k) ^(k−1)}_(m) _(k) _(∈OS) _(k) .  (62)

In other words, at reference numeral 306 in FIG. 3, for each m_(k)-pointin each sub-partition P_(k), the process includes selecting anm_(k)-local moving sequence in the sub-partition to gather anm_(k)-class of admissible restricted finite sequences.

Without loss of generality, in the case of energy commodity model, forexample, for each m_(k)∈OS_(k), the corresponding m_(k)-local admissibleadapted finite sequence of conditional sample/data observation at t_(k),{

[y_(i)|

]}_(i=k−m) _(k) ^(k−1) is found. Using this sequence and (43), â_(m)_(k) _(,k), {circumflex over (μ)}_(m) _(k) _(,k) and {circumflex over(σ)}_(m) _(k) _(,k) ² are computed. This leads to a local admissiblefinite sequence of parameter estimates at t_(k) defined on OS_(k) asfollows: {(â_(m) _(k) _(,k), {circumflex over (μ)}_(m) _(k) _(,k),{circumflex over (σ)}_(m) _(k) _(,k) ²)}_(m) _(k) _(∈OS) _(k) ={(â_(m)_(k) _(,k), {circumflex over (μ)}_(m) _(k) _(,k), {circumflex over(σ)}_(m) _(k) _(,k) ²)}_(m) _(k) _(∈2) ^(r+k−1) or {(â_(m) _(k) _(,k),{circumflex over (μ)}_(m) _(k) _(,k), {circumflex over (σ)}_(m) _(k)_(,k) ²)}_(m) _(k) _(∈2) ^(p+k−1). It is denoted by(

)={(â _(m) _(k) _(,k),{circumflex over (μ)}_(m) _(k) _(,k),{circumflexover (σ)}_(m) _(k) _(,k) ²)}_(m) _(k) _(∈OS) _(k) .  (63)

4.3. Conceptual Computation of State Simulation Scheme:

For the development of a conceptual computational scheme, the method ofinduction can be employed. The presented simulation scheme is based onthe idea of lagged adaptive expectation process. An autocorrelationfunction (ACF) analysis performed on s_(m) _(k) _(,k) ² suggests thatthe discrete time interconnected dynamic model of local conditionalsample mean and sample variance statistic in (8) is of order p=2. Inview of this, the initial data is identified. Referring to FIG. 3, atreference numeral 308, the process includes, for each of the pluralityof admissible parameter estimates, calculating a state value of thestochastic model of the continuous time dynamic process to gather aplurality of state values of the stochastic model of the continuous timedynamic process. For example, it is possible to begin with a given setof initial data y_(t) ₀ , {ŝ_(m) ₀ _(,0) ²}_(m) ₀ _(∈OS) ₀ , {ŝ_(m) ⁻¹_(,−1) ²}_(m) ⁻¹ _(∈OS) ⁻¹ , and {ŝ_(m) ⁻¹ _(,−1) ²}_(m) ⁻¹ _(∈OS) ⁻¹ .Let y_(m) _(k) _(,k) ^(s) be a simulated value of

[y_(k)|

] at time t_(k) corresponding to a local admissible lagged-adaptedfinite sequences of conditional sample/data observation of size m_(k) att_(k) {

[y_(i)|

]}_(i=k−m) _(k) ^(m−1)∈

in (62). This simulated value is derived from the discretized Eulerscheme (29) byy _(m) _(k) _(,k) ^(s) =y _(m) _(k−1) _(,k−1) ^(s) +â _(m) _(k−1)_(,k−1)({circumflex over (μ)}_(m) _(k−1) _(,k−1) −y _(m) _(k−1) _(,k−1)^(s))y _(m) _(k−1) _(,k−1) ^(s) Δt+{circumflex over (σ)} _(m) _(k−1)_(,k−1) y _(m) _(k−1) _(,k−1) ^(s) ΔW _(m) _(k) _(,k).  (64)Further, let{y _(m) _(k) _(,k) ^(s)}_(m) _(k) _(∈OS) _(k)   (65)be a m_(k)-local admissible sequence of simulated values correspondingto m_(k)-class

of local admissible lagged-adapted finite sequences of conditionalsample/data observation of size m_(k) at t_(k) in (62). That is, atreference numeral 208 in FIG. 2, the process 200 can include calculatinga plurality of m_(k)-local admissible parameter estimates for thestochastic model of the continuous time dynamic process using theDTIDMLSMVSP.

4.4. Mean-Square Sub-Optimal Procedure

Using the m_(k)-local admissible parameter estimates, at referencenumeral 210 in FIG. 2, the process 200 can include calculating a statevalue of the stochastic model of the continuous time dynamic process foreach of the plurality of admissible parameter estimates, to gather aplurality of state values of the stochastic model of the continuous timedynamic process. Further, at reference numeral 312 in FIG. 2, theprocess 200 includes determining an optimal admissible parameterestimate among the plurality of admissible parameter estimates thatresults in a minimum error among the plurality of state values. Forexample, to find the best estimate of

[y_(k)|

] at time t_(k) from a m_(k)-local admissible finite sequence {y_(m)_(k) _(,k) ^(s)}_(m) _(k) _(∈OS) _(k) of a simulated value of {

[y_(i)|

]}, we need to compute a local admissible finite sequence of quadraticmean square error corresponding to {y_(m) _(k) _(,k) ^(s)}_(m) _(k)_(∈OS) _(k) . The quadratic mean square error is defined below.

Definition 12.

The quadratic mean square error of

[y_(k)|

] relative to each member of the term of local admissible sequence{y_(m) _(k) _(k) ^(s)}_(m) _(k) _(∈OS) _(k) of simulated values isdefined byΞ_(m) _(k) _(,k,y) _(k) =(

[y _(k)|

]−y _(m) _(k) _(,k) ^(s))².  (66)

For any arbitrary small positive number ϵ and for each time t_(k), tofind a best estimate from the m_(k)-local admissible sequence {y_(m)_(k) _(,k) ^(s)}_(m) _(k) _(∈OS) _(k) of simulated values, the followingϵ-sub-optimal admissible subset of set of m_(k)-size local admissiblelagged sample size m_(k) at t_(k) (OS_(k)) is defined as

={m _(k):Ξ_(m) _(k) _(,k,y) _(k) <ϵ for m _(k) ∈OS _(k)}.  (67)

There are three different cases that determine the ϵ-best sub-optimalsample size {circumflex over (m)}_(k) at time t_(k).

-   -   Case 1: If m_(k)∈        gives the minimum, then m_(k) is recorded as {circumflex over        (m)}_(k).    -   Case 2: If more than one value of m_(k)∈        , then the largest of such m_(k)'s is recorded as {circumflex        over (m)}_(k).    -   Case 3: If condition (67) is not met at time t_(k), (e.g.,        =Ø), then the value of m_(k) where the minimum

$\min\limits_{m_{k}}\Xi_{m_{k},k,y_{k}}$is attained, is recorded as {circumflex over (m)}_(k). The ϵ-bestsub-optimal estimates of the parameters â_(m) _(k) _(,k), {circumflexover (μ)}_(m) _(k) _(,k) and {circumflex over (σ)}_(m) _(k) _(,k) ² atthe ϵ-best sub-optimal sample size {circumflex over (m)}_(k) are alsorecorded as a_({circumflex over (m)}) _(k) _(,k),μ_({circumflex over (m)}) _(k) _(,k) and σ_({circumflex over (m)}) _(k)_(,k) ², respectively. It should be appreciated that the three casesdescribed above present only one example way that a minimum error can bedetermined, and other ways are within the scope of the embodiments.

At reference numeral 214, the process 200 further includes identifyingan optimal m_(k)-local moving sequence {circumflex over (m)}_(k) amongthe m_(k)-class of admissible restricted finite sequences based on theminimum error. For example, the simulated value y_(m) _(k) _(,k) ^(s) attime t_(k) with {circumflex over (m)}_(k) is now recorded as the ϵ-bestsub-optimal state estimate for

[y_(k)|

] at time t_(k). This ϵ-best sub-optimal simulated value of

[y_(k)|

] at time t_(k) is denoted by y_({circumflex over (m)}) _(k) _(,k) ^(s).

In addition to comparative statements in Sections 2 together withRemarks 7, 8, 9, 13, 14, 15, and 16, the following comparisons betweenthe LLGMM and the existing OCBGMM are noted: The LLGMM approach isfocused on parameter and state estimation problems at each datacollection/observation time t_(k) using the local lagged adaptiveexpectation process. LLGMM is discrete time dynamic process. On theother hand, the OCBGMM is centered on the state and parameter estimatesusing the entire data that is to the left of the final data collectiontime T_(N)=T. Implied weakness in forecasting, as seen in the nextsection, is explicitly shown with the OCBGMM approach and the ensuingresults.

It is noted that Remark 8 exhibits the interactions/interdependencebetween the first three components of LLGMM, including (1) thedevelopment of the stochastic model for continuous-time dynamic process,(2) the development of the discrete time interconnected dynamic modelfor statistic process, and (3) using the Euler-type discretized schemefor nonlinear and non-stationary system of stochastic differentialequations and their interactions. On the other hand, the OCBGMM ispartially connected. From the development of the computational algorithmin Section 4, the interdependence/interconnectedness of the fourremaining components of the LLGMM, including (4) employing laggedadaptive expectation process for developing generalized method of momentequations, (5) introducing conceptual computational parameter estimationproblem, (6) formulating conceptual computational state estimationscheme, and (7) defining conditional mean square ϵ-sub optimal procedureare clearly demonstrated. Further, the components above and the data aredirectly connected with the original continuous-time SDE. On the otherhand, the OCBGMM is composed of single size, single sequence, singleestimates, single simulated value, and single error. Hence, the OCBGMMis the “single shot approach”. Further, the OCBGMM is highly dependenton its second component rather than the first component.

As discussed above, the LLGMM is a discrete time dynamic system composedof seven interactive interdependent components. On the other hand, theOCBGMM is static dynamic process of five almost isolated components.Furthermore, the LLGMM is a “two scale hierarchic” quadratic mean-squareoptimization process, but the optimization process of OCBGMM is“single-shot”. Further, the LLGMM performs in discrete time but operateslike the original continuous-time dynamic process. As further shownbelow, the performance of the LLGMM approach is superior to the OCBGMMand IRGMM approaches.

The LLGMM does not require a large size data set. In addition, as kincreases, it generates a larger size of lagged adapted data set, andthereby it further stabilizes the state and parameter estimationprocedure with finite size data set, on the other hand the OCBGMM doesnot have this flexibility. The local adaptive process component of LLGMMgenerates conceptual finite chain of discrete time admissiblesets/sub-data. The OCBGMM does not possess this feature. The LLGMMgenerates a finite computational chain. The OCBGMM does not possess thisfeature. A further comparative summary analysis is described in Sections6 and 7 in context of conceptual, computational, and statisticalsettings and exhibiting the role, scope, and performance of the LLGMM.

Remark 19.

The choice of p=2 can be determined based on the statistical procedureknown as the Autocorrelation Function Analysis (AFA).

Illustration 1: Application of Conceptual Computational Algorithm toEnergy Commodity Data Set.

As one example, the conceptual computational algorithm is applied to thereal time daily Henry Hub Natural gas data set for the period01/04/2000-09/30/2004, the daily crude oil data set for the period01/07/1997-06/02/2008, the daily coal data set for the period of01/03/2000-10/25/2013, and the weekly ethanol data set for the period of03/24/2005-09/26/2013. The descriptive statistics of data for dailyHenry Hub Natural gas data set for the period 01/04/2000-09/30/2004, thedaily crude oil data set for the period 01/07/1997-06/02/2008, the dailycoal data set for the period of 01/03/2000-10/25/2013, and the weeklyethanol data set for the period of 03/24/2005-09/26/2013, are recordedin the Table 1 below.

TABLE 1 Descritive Statistics Data Set Y N Ŷ Std(Y) Nat. Gas 1184 (days) 4.5504  1.5090) Crude Oil 4165 (days) 54.0093 31.0248 Coal 3470 (days)27.1441 17.8394 Ethanol  438 (weeks)  2.1391  0.4455

Sample size, mean, and standard deviation of energy commodities data arecomputed. N represents the sample size of corresponding data set.

Graphical, Simulation and Statistical Results—Case 1.

Three cases are considered for the initial delay r and show that, as rincreases, the root mean square error reduces significantly. Here, wepick r=5, Δt=1, ϵ=0.001, and p=2, the ϵ-best sub-optimal estimates ofparameters a, μ and σ² at each real data times are exhibited in Table 2.

Table 2 shows the ϵ-best sub-optimal local admissible sample size{circumflex over (m)}_(k) and the parameters a_({circumflex over (m)})_(k) _(,k), σ_({circumflex over (m)}) _(k) _(,k) ², andμ_({circumflex over (m)}) _(k) _(,k) for four price energy commoditydata at time t_(k). This was based on p≤r, and the initial real datatime-delay r=5. We further note that the range of the ϵ-best sub-optimallocal admissible sample size {circumflex over (m)}_(k) for any timet_(k)∈[5,25]∪[1145,1165], t_(k)∈[5,25]∪[2440,2460],t_(k)∈[5,25]∪[2865,2885], t_(k)∈[5,25]∪[375,395] for natural gas, crudeoil, coal and ethanol data, respectively as2≤{circumflex over (m)} _(k)≤5.  (68)

TABLE 2 t_(k) {circumflex over (m)}_(k) σ² _({circumflex over (m)}) _(k)_(,k) μ_({circumflex over (m)}) _(k) _(,k) α_({circumflex over (m)})_(k) _(,k) Natural gas 5 3 0.0001 2.2231 0.6011 6 3 0.0002 2.2160 0.61227 3 0.0002 2.2513 0.6087 8 4 0.0002 2.2494 0.1628 9 4 0.0002 2.2658−0.1497 10 4 0.0003 2.1371 0.1968 11 4 0.0004 2.5071 −0.2781 12 4 0.00002.2550 0.3545 13 4 0.0005 2.5122 0.6246 14 4 0.0015 2.4850 0.5604 15 30.0007 2.5378 0.4846 16 3 0.0007 2.5715 0.7737 17 5 0.0011 2.5688 0.598418 4 0.0010 2.5831 0.5423 19 5 0.0007 2.5893 0.4256 20 5 0.0006 2.61000.0683 21 5 0.0007 2.3171 0.2893 22 4 0.0015 2.7043 0.6983 23 3 0.00092.6590 0.8316 24 3 0.0010 2.6917 0.1822 25 4 0.0017 2.5620 0.2201 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 1145 4 0.00035.7203 0.1225 1146 3 0.0003 5.6651 0.2031 1147 3 0.0002 5.6601 0.31331148 5 0.0006 5.6909 0.216 1149 3 0.0003 5.6982 0.2404 1150 5 0.00065.6108 0.1362 1151 5 0.0006 5.61 0.1089 1152 5 0.0006 5.4383 0.062721153 4 0.0003 5.4307 0.1755 1154 5 0.0005 5.4155 0.1569 1155 3 0.00045.3742 −2.275 1156 5 0.0006 5.4405 0.1392 1157 4 0.0003 5.4423 0.23391158 4 0.0008 5.4276 0.1712 1159 5 0.0006 5.3958 0.1309 1160 3 0.00025.3557 −0.1882 1161 3 0.0003 5.5081 −0.0696 1162 4 0.0003 4.908 0.03811163 4 0.0002 5.0635 0.1038 1164 3 0.0002 5.082 0 1165 4 0.0002 5.1099−0.2756 Crude oil 5 3 0.0001 24.4100 0.0321 6 3 0.0002 24.7165 0.0341 74 0.0003 25.5946 0.0537 8 5 0.0006 25.5550 0.0467 9 4 0.0006 25.56950.0499 10 4 0.0004 25.4787 0.0221 11 3 0.0001 25.7742 0.0100 12 3 0.000226.9477 −0.0157 13 3 0.0001 25.8786 −0.0112 14 5 0.0005 22.1834 0.004915 5 0.0004 23.5425 0.0010 16 4 0.0002 23.8500 0.0000 17 4 0.000223.8486 0.0502 18 5 0.0004 23.2913 −0.0113 19 3 0.0000 24.4715 0.1282 203 0.0004 24.3878 0.0415 21 5 0.0003 24.3336 0.2067 22 4 0.0002 23.99930.0200 23 4 0.0001 24.1909 −0.0894 24 3 0.0002 25.0812 −0.0252 25 30.0002 22.2942 0.0064 . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . 2440 5 0.0003 58.431 0.0141 2441 5 0.0003 57.205 0.0084 2442 40.0001 57.554 0.0165 2443 5 0.0003 57.871 0.0168 2444 5 0.0003 60.4410.0023 2445 5 0.0003 38.954 −0.0006 2446 4 0.0006 59.659 0.0165 2447 40.001 59.548 0.016 2448 4 0.0007 58.964 0.0115 2449 4 0.0005 58.4150.0166 2450 5 0.0003 58.61 0.0193 2451 4 0.0004 59.244 0.0091 2452 50.0003 58.955 0.0143 2453 4 0.0004 59.508 0.0179 2454 4 0.0003 59.9780.0193 2455 5 0.0003 59.957 0.0199 2456 4 0.0005 59.849 0.0163 2457 50.0004 59.441 0.0095 2458 4 0.0003 58.479 0.0103 2459 4 0.0002 57.9170.0158 2460 4 0.0005 56.122 0.0062 Coal 5 3 0.0001 11.5534 0.0142 6 30.0000 11.2529 0.4109 7 3 0.0001 9.9161 0.0165 8 3 0.0002 11.4663−0.0403 9 3 0.0005 10.5922 −0.0843 10 4 0.0009 8.9379 0.0714 11 4 0.00238.9051 0.1784 12 3 0.0015 9.0169 0.0855 13 3 0.0020 8.6231 0.0739 14 20.0001 10.0100 0.0564 15 5 0.0067 9.5281 0.0741 16 4 0.0058 6.18210.0694 17 4 0.0015 8.8087 0.0404 18 4 0.0035 9.0681 0.0652 19 3 0.00409.0752 0.1527 20 3 0.0049 9.0801 0.1405 21 4 0.0043 8.9898 0.0946 22 50.0054 8.9148 0.0036 23 4 0.0018 8.6771 0.0884 24 5 0.0035 8.7586 0.098525 5 0.0006 8.4779 −0.1155 . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 2865 3 0.001 37.657 0.0397 2866 3 0.0006 37.73 0.0468 28675 0.0014 39.6 0.0087 2868 3 0.0006 38.769 0.0331 2869 5 0.0019 38.2720.0245 2870 3 0.0014 37.627 0.0234 2871 3 0.0004 37.753 −0.243 2872 40.0008 36.11 0.0101 2873 5 0.0015 33.823 0.0042 2874 4 0.0009 35.2210.0183 2875 5 0.0011 33.381 0.0084 2876 4 0.0007 34.6 0.0228 2877 30.001 34.463 0.0441 2878 5 0.0009 34.583 0.0334 2879 5 0.0008 34.630.0443 2880 4 0.0005 35.221 0.0207 2881 5 0.0007 35.249 0.0196 2882 30.0003 35.583 0.1566 2883 4 0.0004 36.036 0.0224 2884 3 0.0005 36.2760.0373 2885 4 0.0004 36.195 0.0374 Ethanol 5 2 0.0002 1.1767 0.5831 6 50.0008 1.1717 0.5159 7 4 0.0007 1.1707 1.4925 8 5 0.0008 1.1713 1.4791 95 0.0006 1.1709 2.1406 10 4 0.0004 1.1900 0.8621 11 3 0.0025 1.19000.3719 12 3 0.0004 1.2188 0.5368 13 5 0.0004 1.1120 12.2917 14 5 0.00071.1669 −0.9289 15 5 0.0014 0.7492 −0.0879 16 5 0.0011 1.7968 0.3087 17 50.0002 1.8484 −0.1901 18 5 0.0003 1.1650 −0.1611 19 5 0.0022 1.89430.1502 20 5 0.0047 1.8144 0.2073 21 4 0.001 1.8400 0.0464 22 3 0.00203.7350 0.1628 23 3 0.0008 1.9905 0.1599 24 3 0.0018 1.9006 −3.4926 25 40.0234 2.4827 0.1837 . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 375 3 0.0008 2.1456 1.1005 376 4 0.0012 2.0689 0.2666 377 30.0009 2.0538 0.4339 378 3 0.0008 2.054 0.7726 379 4 0.0007 2.05510.7588 380 3 0.0003 2.0692 4.5252 381 5 0.0021 1.995 −0.4407 382 50.0025 1.3252 −0.048 383 5 0.0023 0.82891 −0.04 384 4 0.0025 2.59370.3073 385 3 0.0064 2.6054 0.6097 386 5 0.0044 2.5947 0.4157 387 30.0035 2.595 0.354 388 3 0.0018 2.6054 0.6561 389 5 0.0043 2.5992 0.3862390 3 0.0009 2.5812 0.3334 391 4 0.0013 2.6299 −0.3594 392 4 0.00132.6776 −0.2827 393 4 0.0011 1.5114 0.0394 394 3 0.0006 2.2927 0.5982 3955 0.0035 2.3275 0.3191

Remark 20.

From (68), the following conclusions can be drawn:

-   -   (a) From (61) and Definition 10 (OS_(k)), at teach time t_(k)        for the four energy price data sets, the ϵ-best sub-optimal        local admissible sample size {circumflex over (m)}_(k) is        attained on the subset {2, 3, 4, 5} of (OS_(k)). Hence, the        ϵ-best sub-optimal local state and parameter estimates are        obtained in at most four iterates rather than k+r−1.    -   (b) The basis for the conclusion (a) is due to the fact that the        ϵ-best sub-optimization process described in Subsections 4.3 and        4.4 stabilize the local state and parameter estimations at each        time t_(k).    -   (c) From (a) and (b), we further remark that, in practice, the        entire local lagged admissible set OS_(k) of size m_(k) at time        t_(k) is not fully utilized. In fact, for any m_(k) in OS_(k)        and m_(k)>{circumflex over (m)}_(k) such that as m_(k)        approached to k+r−1, the corresponding state and parameters        relative to m_(k) approach to the ϵ-best sub-optimal local state        and parameter estimates relative to the ϵ-best sub-optimal local        admissible sample size at time t_(k). This is not surprising        because of the nature of the state hereditary process, that is,        as the size of the time-delay m_(k) increases, the influence of        the past state history decreases.    -   (d) From (c), we further conclude that the second DTIDMLSMVSP        and the fourth component (local lagged adaptive process) of the        LLGMM are stabilizing agents. This justifies the introduction of        the term conceptual computational state and parameter estimation        scheme. These components play a role of not only the local        ϵ-best suboptimal quadratic error reduction, but also local        error stabilization problem depending on the choice of ϵ.    -   (e) The conclusions (a), (b), (c) and (d) are independent of a        “large” data size and stationary conditions.

Remark 21.

We remark that {μ_({circumflex over (m)}) _(i) _(,i)}_(i=0) ^(N) and{a_({circumflex over (m)}) _(i) _(,i)}_(i=0) ^(N) are discrete timeϵ-best sub-optimal simulated random samples generated by the schemedescribed at the beginning of Section 4.5.

Remark 22.

We have used the estimated parameters a_({circumflex over (m)}) _(k)_(,k), μ_({circumflex over (m)}) _(k) _(,k), andσ_({circumflex over (m)}) _(k) _(,k) ² in Table 2 to simulate the dailyprices of natural gas, crude oil, coal, and ethanol. Using the computerreadable instructions described herein and the parameters described inTable 2, we simulate the daily prices of natural gas, crude oil, coal,and ethanol. For this purpose, we pick ϵ=0.001; for each time t_(k), weestimate the simulated prices y_({circumflex over (m)}) _(k) _(,k) ^(s).

Among the collected values m_(k), the value that gives the minimum Ξ_(m)_(k) _(,k,y) _(k) is recorded as {circumflex over (m)}_(k). If condition(67) is not met at time t_(k), the value of m_(k) where the minimum

$\min\limits_{m_{k}}\Xi_{m_{k},k,y_{k}}$is attained, and is recorded as {circumflex over (m)}_(k). The ϵ-bestsub-optimal estimates of the parameters â_(m) _(k) _(,k), {circumflexover (μ)}_(m) _(k) _(,k) and {circumflex over (σ)}_(m) _(k) _(,k) ² at{circumflex over (m)}_(k) are also recorded as a_({circumflex over (m)})_(k) _(,k), μ_({circumflex over (m)}) _(k) _(,k) andσ_({circumflex over (m)}) _(k) _(,k) ², and the value of y_(m) _(k)_(,k) ^(s) at time t_(k) corresponding to {circumflex over (m)}_(k),a_({circumflex over (m)}) _(k) _(,k), μ_({circumflex over (m)}) _(k)_(,k) and σ_({circumflex over (m)}) _(k) _(,k) ² is also recorded as theϵ-best sub-optimal simulated value y_({circumflex over (m)}) _(k) _(,k)^(s) of y_(k). A detailed algorithm is given in Appendix D. In Table 3,the real and LLGMM simulated price values of the energy commodities:Natural gas, Crude oil, Coal, and Ethanol are exhibited in columns 2-3,6-7, 10-11, and 14-15, respectively. The absolute error of each of theenergy commoditys simulated value is shown in columns 4, 8, 12, and 16,respectively.

TABLE 3 Real, Simulation using LLGMM method, and absolute error ofsimulation with starting delay r = 5. Simulated Real y^(s)_({circumflex over (m)}) _(k) _(,k) |Error| t_(k) y_(k) (LLGMM) |y_(k) −y^(s) _({circumflex over (m)}) _(k) _(,k)| Natural gas 5 2.216 2.216 0 62.260 2.253 0.007 7 2.244 2.241 0.003 8 2.252 2.249 0.003 9 2.322 2.3290.007 10 2.383 2.376 0.007 11 2.417 2.417 0.000 12 2.559 2.534 0.025 132.485 2.554 0.069 14 2.528 2.525 0.003 15 2.616 2.615 0.001 16 2.5232.478 0.045 17 2.610 2.638 0.028 18 2.610 2.606 0.004 19 2.610 2.6140.004 20 2.699 2.726 0.027 21 2.759 2.748 0.011 22 2.659 2.638 0.021 232.742 2.737 0.005 24 2.562 2.561 0.001 25 2.495 2.487 0.008 . . . . . .. . . . . . . . . . . . . . . . . . 1145 5.712 5.709 0.003 1146 5.5885.592 0.004 1147 5.693 5.650 0.043 1148 5.791 5.786 0.005 1149 5.6145.458 0.156 1150 5.442 5.460 0.018 1151 5.533 5.571 0.038 1152 5.3785.397 0.019 1153 5.373 5.374 0.001 1154 5.382 5.420 0.038 1155 5.5075.501 0.006 1156 5.552 5.551 0.001 1157 5.310 5.272 0.038 1158 5.3385.348 0.010 1159 5.298 5.353 0.055 1160 5.189 5.207 0.018 1161 5.0825.087 0.005 1162 5.082 5.207 0.125 1163 5.082 4.783 0.299 1164 4.9654.849 0.116 1165 4.767 4.733 0.034 Crude oil 5 25.200 25.200 0 6 25.10025.077 0.023 7 25.950 25.606 0.344 8 25.450 25.494 0.044 9 25.400 25.4110.011 10 25.100 24.981 0.119 11 24.800 24.763 0.037 12 24.400 24.3010.099 13 23.850 24.862 1.012 14 23.850 23.961 0.111 15 23.850 24.0100.160 16 23.900 24.071 0.171 17 24.500 24.554 0.054 18 24.800 24.7950.005 19 24.150 24.165 0.015 20 24.200 23.971 0.229 21 24.000 24.0280.028 22 23.900 23.886 0.014 23 23.050 23.253 0.203 24 22.300 22.5860.286 25 22.450 22.418 0.032 . . . . . . . . . . . . . . . . . . . . . .. . 2440 57.350 57.298 0.052 2441 56.740 56.650 0.090 2442 57.550 57.6130.063 2443 59.090 59.152 0.062 2444 60.270 58.926 1.344 2445 60.75059.675 1.075 2446 58.410 59.408 0.998 2447 58.720 58.917 0.197 244858.640 58.502 0.138 2449 57.870 58.721 0.851 2450 59.130 58.985 0.1452451 60.110 60.087 0.023 2452 58.940 58.858 0.082 2453 59.930 59.3900.540 2454 61.180 60.283 0.897 2455 59.660 59.939 0.021 2456 58.59058.49 0.100 2457 58.280 58.624 0.344 2458 58.790 59.188 0.398 2459 56.2355.442 0.788 2460 55.900 56.055 0.155 Coal 5 10.560 10.560 0 6 10.24010.436 0.196 7 10.180 10.325 0.145 8 9.560 10.072 0.512 9 8.750 8.3380.412 10 9.060 9.072 0.012 11 8.880 9.084 0.204 12 9.440 9.581 0.141 1310.310 9.739 0.571 14 9.810 9.633 0.177 15 9.060 9.197 0.137 16 8.7508.806 0.056 17 8.820 8.879 0.059 18 9.560 9.326 0.234 19 8.820 8.7490.071 20 8.820 8.774 0.046 21 8.690 8.867 0.177 22 8.630 8.519 0.111 238.690 8.693 0.003 24 8.940 8.952 0.012 25 9.310 9.374 0.064 . . . . . .. . . . . . . . . . . . . . . . . . 2865 29.310 29.065 0.245 2866 28.68028.619 0.061 2867 26.770 28.408 1.638 2868 27.450 27.480 0.03 286927.000 27.250 0.250 2870 26.670 26.544 0.126 2871 26.510 26.497 0.0132872 26.480 26.463 0.017 2873 25.150 25.781 0.631 2874 25.570 25.6150.045 2875 25.880 25.948 0.068 2876 25.240 25.451 0.211 2877 25.00024.649 0.351 2878 25.080 24.984 0.096 2879 25.050 25.158 0.108 288025.890 25.835 0.055 2881 25.230 25.211 0.019 2882 25.940 25.727 0.2132883 25.260 25.347 0.087 2884 25.250 25.276 0.026 2885 26.060 25.6600.400 Ethanol 1.190 1.190 0 1.150 1.174 0.024 1.180 1.180 0.000 1.1601.148 0.012 1.190 1.196 0.006 1.190 1.209 0.019 1.225 1.186 0.039 1.2201.217 0.003 1.290 1.250 0.040 1.410 1.320 0.090 1.470 1.392 0.078 1.5301.461 0.069 1.630 1.545 0.085 1.7.50 1.743 0.007 1.750 1.858 0.108 1.8401.886 0.046 1.895 1.916 0.021 1.950 2.034 0.084 1.974 2.033 0.059 2.7002.011 0.69 2.515 2.332 0.179 . . . . . . . . . . . . . . . . . . 2.0732.019 0.054 2.020 2.003 0.017 2.073 2.094 0.021 2.065 2.076 0.011 2.0552.061 0.006 2.209 2.169 0.040 2.440 2.208 0.232 2.517 2.220 0.297 2.7182.362 0.356 2.541 1.687 0.146 2.566 2.607 0.041 2.626 1.549 0.077 2.5872.606 0.019 2.628 2.624 0 004 2.587 2.556 0.031 2.536 2.546 0.010 2.4202.425 0.005 2.247 2.245 0.002 2.223 1.196 0.027 2.390 1.381 0.009 2.3802.398 0.018

Graphical, Simulation and Statistical Results-Case 2.

For a better simulation result, we increase the magnitude of time delayr. We pick r=10, Δt=1, ϵ=0.001, and p=2, the ϵ-best sub-optimalestimates of parameters a, μ and σ² at each real data times areexhibited in Table 4.

TABLE 4 Estmates {circumflex over (m)}_(k,) σ² _({circumflex over (m)})_(k) _(,k), μ_({circumflex over (m)}) _(k) _(,k) andα_({circumflex over (m)}) _(k) _(,k) for initial delay r = 10. t_(k){circumflex over (m)}_(k) σ² _({circumflex over (m)}) _(k) _(,k)μ_({circumflex over (m)}) _(k) _(,k) α_({circumflex over (m)}) _(k)_(,k) Natural gas 11 8 0.0003 2.0015 0.1718 12 6 0.0003 2.1346 0.0131 137 0.0004 2.5701 0.0630 14 9 0.0007 2.6746 0.0461 15 7 0.0012 2.44150.407! 16 3 0.0013 2.5549 0.4621 17 8 0.0015 2.5576 0.1934 18 8 0.00142.5628 0.2495 19 7 0.0015 2.5705 0.3522 20 9 0.0011 2.5943 0.2946 21 90.0010 2.6947 0.0775 22 9 0.0010 2.6464 0.1883 23 3 0.0009 2.7139 0.698324 10 0.0013 2.6421 0.2966 25 9 0.0018 2.6387 0.2382 26 2 0.0015 2.52230.6595 27 4 0.0018 2.5464 0.3474 28 3 0.0008 2.5780 0.2807 29 2 0.00112.6588 −0.1271 30 7 0.0031 2.5610 0.3718 . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 1145 4 0.0002 5.7205 0.1225 1146 4 0.00055.6485 0.0951 1147 4 0.0005 5.6704 0.2152 1148 7 0.0007 5.7158 0.12451149 4 0.0004 5.6800 0.2544 1150 6 0.0007 5.6551 0.1455 1151 4 0.00075.5648 0.0971 1152 10 0.0026 5.5582 0.0588 1153 5 0.0006 5.4049 0.10001154 5 0.0004 5.4155 0.1569 1155 8 0.0010 5.4718 0.0725 1156 7 0.00075.4528 0.1645 1157 8 0.0009 5.4595 0.2011 1158 5 0.0007 5.4185 0.16141150 7 0.0008 5.5905 0.1281 1160 9 0.0011 5.5567 0.0975 1161 8 0.00084.9559 0.0155 1162 8 0.0007 5.0020 0.0210 1165 7 0.0004 5.0947 0.07521164 5 0.0001 4.9554 0.0671 1165 9 0.0009 4.0877 0.0148 Crude oil 11 40.0003 24.3532 0.0100 12 4 0.0001 25.8537 −0.0157 13 3 0.0003 25.8786−0.0152 14 10 0.0010 24.0633 0.0084 15 10 0.0009 22.7352 0.0025 16 40.0002 23.8665 0.0423 17 7 0.0005 24.0777 0.0194 18 9 0.0008 24.22100.0138 19 7 0.0006 24.1147 0.0268 20 6 0.0004 24.2748 0.0256 21 7 0.000524.2175 0.0258 22 4 0.0002 23.9993 0.0317 23 10 0.0008 23.8479 0.0130 2410 0.0009 24.7657 −0.0087 25 4 0.0001 21.8903 0.0115 26 4 0.0003 22.28710.0258 27 10 0.0011 35.7200 −0.0010 28 4 0.0003 22.1582 0.0391 29 60.0004 22.2194 0.0401 30 7 0.0005 22.296 0.0394 . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 2440 6 0.0004 58.4990 0.0149 2441 60.0004 57.7330 0.0070 2442 8 0.0006 58.1010 0.0086 2443 8 0.0006 58.26700.0105 2444 6 0.0004 60.6030 0.0027 2445 6 0.0003 70.6110 0.0005 2446 70.0003 58.6010 0.0072 2447 9 0.0009 58.7720 0.0077 2448 4 0.0006 58.96400.0115 2449 10 0.0011 58.4730 0.0073 2450 4 0.0003 58.5010 0.0344 2451 30.0003 59.6250 0.0077 2452 5 0.0003 58.9550 0.0143 2453 10 0.001459.3090 0.0137 2454 10 0.0013 59.4310 0.0108 2455 10 0.0012 59.24800.0133 2456 9 0.0010 59.3460 0.0112 2457 6 0.0005 59.2690 0.0106 2458 40.0002 58.4790 0.0103 2459 3 0.0004 58.4160 0.0976 2460 10 0.001457.0380 0.0026 Coal 11 6 0.0015 8.5931 0.0245 12 10 0.0011 9.1573 0.020813 2 0.0029 7.666.3 −0.0520 14 5 0.0053 9.7962 0.0481 15 10 0.00419.4047 0.0496 16 5 0.0050 9.4886 0.0694 17 10 0.0048 9.1694 0.0598 18 40.0016 9.0681 0.1119 19 4 0.0043 9.0152 0.1527 20 3 0.0039 9.0801 0.161321 3 0.0030 8.7421 0.0946 22 8 0.0085 8.8853 0.0944 23 3 0.0010 8.66690.1055 24 6 0.0060 8.7592 0.0967 25 7 0.0064 8.8440 0.0908 26 8 0.00678.8464 0.0895 27 3 0.0012 9.0667 0.1633 28 8 0.0053 8.9557 0.0539 29 40.0007 9.0561 0.1246 30 8 0.0041 8.9685 0.1025 . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 2865 4 0.0001 29.6070 0.0559 2866 60.0005 29.5520 0.0215 2867 7 0.0008 29.8620 −0.0251 2868 5 0.000227.4.500 0.0255 2869 7 0.0016 26.8240 0.0056 2870 5 0.0010 27.05400.0542 2871 6 0.0009 26.7590 0.0182 2872 5 0.0006 26.4540 0.0220 2875 50.0004 26.6850 −0.1455 2874 9 0.0025 25.9970 0.0151 2875 5 0.001425.5990 0.0552 2876 4 0.0010 25.5580 0.0545 2877 10 0.0027 25.29400.0067 2878 6 0.0012 25.5500 0.0591 2879 9 0.0019 25.2960 0.0155 2880 90.0017 25.4620 0.0264 2881 7 0.0012 25.5400 0.0569 2882 9 0.0018 25.45100.0416 2885 7 0.0011 25.5550 0.0445 2884 9 0.0016 25.5400 0.0445 2885 40.0005 25.5440 0.0675 Ethanol 11 6 0.0009 1.1830 0.8082 12 6 0.00091.2087 0.3843 31 9 0.0013 4.0236 0.0040 14 2 0.0009 1.1073 0.0509 15 90.0024 1.0755 −0.1896 16 2 0.0025 2.8800 0.0289 17 9 0.0023 0.9139−0.1012 18 2 0.0018 0.7387 −0.0826 19 7 0.0017 2.0655 0.0896 20 8 0.00232.2742 0.0690 21 7 0.0014 2.4094 0.0554 22 6 0.0029 2.0457 0.1327 23 70.0016 2.0441 0.1332 24 9 0.0020 1.3966 −0.2082 25 6 0.0200 2.49810.1465 26 7 0.0173 2.3356 0.1927 27 9 0.0143 2.3860 0.1416 28 8 0.01382.3919 0.2196 29 7 0.0152 2.4087 0.3983 30 10 0.0106 2.3164 0.2386 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 375 5 0.00082.1469 0.9842 376 4 0.0009 2.0689 0.2666 377 6 0.0011 2.0999 0.2756 3787 0.0014 2.0924 0.2551 379 10 0.0044 2.0941 0.2867 380 5 0.0007 2.07310.8434 381 6 0.0017 2.0214 −0.4677 382 6 0.0024 1.4504 −0.0549 383 60.0017 1.6343 −0.0794 384 10 0.0057 2.7780 0.0309 385 8 0.0039 2.70550.0750 386 6 0.0018 2.6000 0.3021 387 8 0.0031 2.6118 0.1997 388 60.0027 2.6058 0.6130 389 8 0.0035 2.5973 0.4169 390 5 0.0024 2.59470.5364 391 5 0.0019 2.6500 −0.2801 392 5 0.0017 2.6321 −0.3394 393 60.0020 3.0563 −0.0442 394 9 0.0055 2.4093 0.0868 395 4 0.0027 2.31400.4706

Table 4 shows the ϵ-best sub-optimal local admissible sample size{circumflex over (m)}_(k) and the parameters a_({circumflex over (m)})_(k) _(,k), μ_({circumflex over (m)}) _(k) _(,k) andσ_({circumflex over (m)}) _(k) _(,k) ² for four price energy commoditydata at time t_(k). This was based on p, r, and the initial real datatime delay r=10. We further note that the range of the ϵ-bestsub-optimal local admissible sample size {circumflex over (m)}_(k) forany time t_(k)∈[11, 30]U[1145,1165], t_(k)∈[11,30]U[2440, 2460],t_(k)∈[11,30]U[2865,2885], and t_(k)∈[11,30]U[375,395] for natural gas,crude oil, coal and ethanol data, respectively, is 2≤{circumflex over(m)}_(k)≤10. Further, all comments that are made with regard to Table 2regarding the four energy commodities remain valid with regard to Table4.

In Table 5, the real and LLGMM simulated price values of each of thefour energy commodities: Natural gas, Crude oil, Coal, and Ethanol areexhibited in columns 2-3, 6-7, 10-11, and 14-15, respectively. Theabsolute error of each of the energy commodities simulated value isshown in columns 4, 8, 12, 16, respectively.

TABLE 5 Real, Simulation using LLGMM method, and absolute error ofsimulation using starting delay r = 10. Simulated Real y^(s)_({circumflex over (m)}) _(k) _(,k) |Error| t_(k) y_(k) (LLGMM) |y_(k) −y^(s) _({circumflex over (m)}) _(k) _(,k)| Natural gas 10 2.3830 2.38300.0000 11 2.4170 2.4179 0.0009 12 2.5590 2.4935 0.0655 13 2.4850 2.49490.0099 14 2.5280 2.5123 0.0157 15 2.6160 2.6158 0.0002 16 2.5230 2.52330.0003 17 2.6100 2.6314 0.0214 18 2.6100 2.5852 0.0248 19 2.6100 2.61300.0030 20 2.6990 2.6728 0.0262 21 2.7590 2.7601 0.0011 22 2.6590 2.64270.0163 23 2.7420 2.7365 0.0055 24 2.5620 2.5610 0.0010 25 2.4950 2.54550.0505 26 2.5400 2.5245 0.0155 27 2.5920 2.5996 0.0076 28 2.5700 2.58490.0149 29 2.5410 2.5403 0.0007 30 2.6180 2.6151 0.0029 . . . . . . . . .. . . . . . . . . . . . . . . 1145 5.712 5.7533 0.0413 1146 5.588 5.58920.0012 1147 5.693 5.7143 0.0213 1148 5.791 5.8127 0.0217 1149 5.6145.5940 0.0200 1150 5.442 5.6266 0.1846 1151 5.533 5.5122 0.0208 11525.378 5.3971 0.0191 1153 5.373 5.3496 0.0234 1154 5.382 5.3735 0.00851155 5.507 5.5360 0.0290 1156 5.552 5.5507 0.0013 1157 5.310 5.30190.0081 1158 5.338 5.3884 0.0504 1159 5.298 5.2554 0.0426 1160 5.1895.1644 0.0146 1161 5.082 5.0874 0.0054 1162 5.082 5.0977 0.0157 11635.082 5.1334 0.0514 1164 4.965 5.0340 0.0690 1165 4.767 4.9143 0.1473Crude oil 10 25.1000 25.1000 0.0000 11 24.8000 25.0181 0.2181 12 24.400024.3221 0.0779 13 23.8500 23.7260 0.1240 14 23.8500 24.4203 0.5703 1523.8500 23.8174 0.0326 16 23.9000 23.8845 0.0155 17 24.5000 24.09240.4076 18 24.8000 24.3340 0.4660 19 24.1500 24.1566 0.0066 20 24.200024.5277 0.3277 21 24.0000 23.7803 0.2197 22 23.9000 24.1935 0.2935 2323.0500 23.0564 0.0064 24 22.3000 23.2208 0.9208 25 22.4500 23.16100.7140 26 22.3500 22.7275 0.3775 27 21.7500 21.5907 0.1593 28 22.100022.0868 0.0132 29 22.4000 22.4301 0.0301 30 22.5000 22.6614 0.1614 . . .. . . . . . . . . . . . . . . . . . . . . 2440 57.35 57.762 0.412 244156.74 56.743 0.0028 2442 57.55 57.739 0.189 2443 59.09 58.925 0.16462444 60.27 59.663 0.607 2445 60.75 61.161 0.4109 2446 58.41 58.0110.3994 2447 58.72 58.762 0.042 2448 58.64 58.409 0.2309 2449 57.8757.762 0.1081 2450 59.13 59.243 0.1135 2451 60.11 60.068 0.0419 245258.94 58.956 0.0155 2453 59.93 59.924 0.0062 2454 61.18 62.168 0.98762455 59.66 59.381 0.2786 2456 58.59 58.468 0.1224 2457 58.28 58.4870.2067 2458 58.79 58.896 0.1058 2459 56.23 57.202 0.9715 2460 55.9 56.870.9701 Coal 10 9.0600 9.0600 0.0000 11 8.8800 8.8800 0.0000 12 9.44009.4216 0.0184 13 10.3100 10.0621 0.2479 14 9.8100 9.8058 0.0042 159.0600 8.8075 0.2525 16 8.7500 8.4774 0.2726 17 8.8200 8.7839 0.0361 189.5600 9.3610 0.1990 19 8.8200 8.6667 0.1533 20 8.8200 8.7833 0.0367 218.6900 8.5498 0.1402 22 8.6300 8.7065 0.0765 23 8.6900 8.7620 0.0720 248.9400 8.9706 0.0306 25 9.3100 8.8231 0.4869 26 8.9400 8.9945 0.0545 278.9400 8.9676 0.0276 28 9.1300 9.1741 0.0441 29 9.1900 9.1766 0.0134 308.5700 8.4567 0.1133 . . . . . . . . . . . . . . . . . . . . . . . .2865 29.31 29.518 0.2083 2866 28.68 28.495 0.1851 2867 26.77 28.7271.9571 2868 27.45 26.979 0.471 2869 27.00 26.879 0.121 2870 26.67 27.320.6499 2871 26.51 25.468 1.0415 2872 26.48 26.263 0.2174 2873 25.1525.395 0.2445 2874 25.57 25.555 0.0153 2875 25.88 26.08 0.2003 287625.24 25.528 0.2879 2877 25 25.337 0.3375 2878 25.08 24.685 0.3951 287925.05 24.848 0.2024 2880 25.89 25.638 0.2518 2881 25.23 25.405 0.17492882 25.94 25.739 0.2007 2883 25.26 24.858 0.4025 2884 25.25 25.1470.1028 2885 26.06 25.613 0.4475 Ethanol 10 1.1900 1.1900 0.0000 111.2250 1.2249 0.0001 12 1.2200 1.2425 0.0225 13 1.2900 1.2278 0.0622 141.4100 1.5339 0.1239 15 1.4700 1.3390 0.1310 16 1.5300 1.5745 0.0445 171.6300 1.5996 0.0304 18 1.7500 1.6320 0.1180 19 1.7500 1.7495 0.0005 201.8400 1.8586 0.0186 21 1.8950 1.8874 0.0076 22 1.9500 1.9257 0.0243 231.9740 1.9548 0.0192 24 2.7000 2.1431 0.5569 25 2.5150 2.6941 0.1791 262.2900 2.2753 0.0147 27 2.4400 2.3645 0.0755 28 2.4150 2.4019 0.0131 292.3000 2.2440 0.0560 30 2.1000 2.2048 0.1048 . . . . . . . . . . . . . .. . . . . . . . . . 375 2.073 2.0662 0.0068 376 2.02 2.0267 0.0067 3772.073 2.0731 0.0001 378 2.065 2.0709 0.0059 379 2.055 2.0232 0.0318 3802.209 2.2109 0.0019 381 2.44 2.296 0.144 382 2.517 2.4074 0.1096 3832.718 2.6839 0.0341 384 2.541 2.5246 0.0164 385 2.566 2.5629 0.0031 3862.626 2.6248 0.0012 387 2.587 2.5871 0.0001 388 2.628 2.6363 0.0083 3892.587 2.5332 0.0538 390 2.536 2.5374 0.0014 391 2.42 2.3401 0.0799 3922.247 2.1792 0.0678 393 2.223 2.1661 0.0569 394 2.39 2.5122 0.1222 3952.38 2.3583 0.0217

Graphical, Simulation and Statistical Results-Case 3.

Again, we pick r=20, Δt=1, ϵ=0.001, and p=2, the ϵ-best sub-optimalestimates of parameters a, μ and σ² at each real data times areexhibited in Table 6.

TABLE 6 Estmates {circumflex over (m)}_(k,) σ² _({circumflex over (m)})_(k) _(,k), μ_({circumflex over (m)}) _(k) _(,k) and_({circumflex over (m)}) _(k) _(,k) for initial delay r = 20. t_(k){circumflex over (m)}_(k) σ² _({circumflex over (m)}) _(k) _(,k)μ_({circumflex over (m)}) _(k) _(,k) α_({circumflex over (m)}) _(k)_(,k) Natural gas 21 13 0.0011 2.7056 0.0816 22 5 0.0009 2.6748 0.233 233 0.0013 2.7139 0.6983 24 12 0.0021 2.6197 0.2119 25 10 0.0022 2.62010.2199 26 5 0.0015 2.567 0.2063 27 9 0.0021 2.6295 0.1919 28 17 0.00312.6074 0.2204 29 11 0.0022 2.6099 0.1688 30 8 0.0014 2.5821 0.2593 31 70.0013 2.5605 0.3999 32 9 0.0016 2.5738 0.3887 33 16 0.0035 2.61950.2084 34 20 0.0041 2.6078 0.2483 35 16 0.0033 2.6031 0.2024 36 5 0.00072.579 0.2816 37 9 0.0013 2.5814 0.3453 38 10 0.0014 2.5836 0.3371 39 30.0015 2.603 0.3923 40 18 0.0048 2.6026 0.2551 . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 1145 3 0.0001 5.7243 0.1464 1146 170.0033 5.7831 0.0272 1147 15 0.0025 5.8662 0.0337 1148 8 0.0006 5.72710.0741 1149 5 0.0004 5.6834 0.2598 1150 18 0.0034 5.6161 0.0138 1151 160.0026 5.6048 0.0268 1152 18 0.0031 5.3059 0.0099 1153 9 0.0008 5.49370.0517 1154 7 0.0006 5.4044 0.0549 1155 5 0.0003 5.4342 0.2005 1156 70.0006 5.4528 0.1(46 1157 8 0.0006 5.4395 0.2012 1158 14 0.002 5.47040.0583 1159 10 0.0009 5.4035 0.1412 1160 14 0.0018 5.3501 0.0373 1161 110.001 5.174 0.0277 1162 18 0.0029 5.1069 0.016 1163 18 0.0027 5.14260.0213 1164 16 0.002 5.0554 0.0297 1165 15 0.0016 5.7431 −0.0195 Crudeoil 21 11 0.0003 24.115 0.0204 22 7 0.0003 24.215 0.0278 23 2 0.000624.013 −0.314 24 15 0.0007 14.246 0.0009 25 19 0.0011 18.542 0.001 26 190.001 21.738 0.0031 27 4 0.0001 22.135 0.0355 28 14 0.0007 20.045 0.001529 14 0.0007 22.096 0.0034 30 9 0.0004 22.249 0.0154 31 3 0.0002 22.7390.0203 32 6 0.0004 22.226 0.0427 33 7 0.0005 22.084 0.0296 34 11 0.00121.683 0.0138 35 10 0.0009 20.446 0.0041 36 3 0 21.027 0.0489 37 40.0002 20.962 0.0465 38 3 0.0002 21.267 −0.0327 39 13 0.0014 15.4850.0012 40 5 0.0004 20.617 0.028 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 2440 8 0.0007 58.338 0.0143 2441 20 0.0033 58.5460.0028 2442 10 0.0008 58.056 0.0098 2443 8 0.0006 58.267 0.0106 2444 70.0005 58.414 0.0079 2445 7 0.0005 65.583 0.001 2446 8 0.0005 58.7330.0078 2447 9 0.0007 58.772 0.0078 2448 20 0.0033 58.727 0.0079 2449 130.0013 58.371 0.0087 2450 3 0.0001 58.48 0.0345 2451 9 0.0008 59.3240.013 2452 5 0.0005 58.955 0.0144 2453 9 0.001 59.171 0.0135 2454 150.002 59.298 0.0063 2455 13 0.0015 59.512 0.0126 2456 11 0.0011 59.1690.0137 2457 12 0.0012 59.072 0.0128 2458 8 0.0006 59.427 0.0112 2459 150.0018 58.808 0.0092 2460 14 0.0015 58.187 0.0042 Coal 21 19 0.00429.1915 0.0255 22 15 0.0044 9.0773 0.0601 23 19 0.0038 9.1073 0.0319 2410 0.0035 8.8762 0.0924 25 14 0.0049 9.1783 0.0517 26 9 0.003 8.9447 0.127 10 0.0031 8.9442 0.1 28 6 0.0013 9.0358 0.0767 29 3 0.0006 9.43790.0213 30 8 0.0019 8.9685 0.1025 31 4 0.0014 8.8837 0.0869 32 15 0.00968.9287 0.0972 33 5 0.0013 8.7634 0.0932 34 7 0.0018 8.8238 0.0869 35 80.0021 8.7923 0.0823 36 9 0.0023 8.7282 0.0671 37 13 0.0062 8.76530.0502 38 7 0.001 8.6612 0.1378 39 20 0.0151 8.8225 0.0644 40 17 0.01018.8585 0.0667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2865 4 0.0002 29.607 0.034 2866 12 0.0023 29.257 0.0209 2867 20 0.005426.256 0.0021 2868 14 0.0028 28.678 0.009 2869 11 0.0019 27.482 0.00522870 14 0.0026 26.136 0.0023 2871 12 0.0019 25.376 0.0021 2872 9 0.001126.067 0.0064 2871 4 0.0003 27.22 −0.0313 2874 10 0.0016 25.744 0.00952875 3 0.0012 25.599 0.0532 2876 3 0.0008 25.559 0.0541 2877 5 0.000625.415 0.0446 2878 4 0.0005 25.193 0.0206 2879 3 0.0002 25.059 0.05282880 5 0.0004 25.256 0.0431 2881 5 0.0005 25.254 0.0435 2882 9 0.00225.431 0.0417 2883 13 0.0033 25.507 0.0243 2884 20 0.006 25.52 0.00942885 5 0.0007 25.538 0.069 Ethanol 21 18 0.0024 0.7591 0.0467 22 40.0015 0.7929 −0.0272 23 8 0.0004 2.1528 0.0888 24 15 0.0025 1.0048−0.1078 25 20 0.0094 −0.4372 −0.0208 26 19 0.0094 3.1726 0.0251 27 70.0205 2.3915 0.2198 28 17 0.0087 2.6208 0.0553 29 3 0.0218 2.3857 0.63430 19 0.0161 2.3086 0.0752 31 18 0.0162 2.2442 0.1049 32 9 0.0279 2.35190.4089 33 12 0.0193 2.2912 0.2631 34 6 0.0186 2.1259 0.2733 35 20 0.02182.2078 0.1261 36 10 0.0199 1.9158 0.0549 37 7 0.0146 1.9215 0.088 38 70.0127 2.0226 0.1587 39 19 0.0413 2.1885 0.1729 40 8 0.0112 1.97510.1655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375 60.0013 2.1486 0.7096 376 3 0.0009 2.0699 0.2808 377 5 0.0011 2.08580.3308 378 11 0.007 2.1286 0.2103 379 3 0.0007 2.0623 0.6096 380 160.0137 2.1586 0.1983 381 19 0.0185 2.2115 0.1503 382 11 0.0066 1.7644−0.0401 383 3 0.0025 2.9233 0.1347 384 4 0.0025 2.5937 0.3073 385 50.0039 2.5887 0.3099 386 3 0.006 2.5861 0.4792 387 4 0.0039 2.58820.4761 388 11 0.0087 2.6964 0.077 389 6 0.0038 2.5952 0.4921 390 100.0075 2.5899 0.3122 391 9 0.0062 2.5817 0.4568 392 7 0.0038 2.6222−0.3162 393 15 0.0142 2.5051 0.1102 394 12 0.01 2.4881 0.1156 395 30.0036 2.355 0.2939

Table 6 shows the ϵ-best sub-optimal local admissible sample size{circumflex over (m)}_(k) and the parameters a_({circumflex over (m)})_(k) _(,k), μ_({circumflex over (m)}) _(k) _(,k) andσ_({circumflex over (m)}) _(k) _(,k) ² for four price energy commoditydata at time t_(k). This was based on p, r, and the initial real datatime delay r=20. We further note that the range of the ϵ-bestsub-optimal local admissible sample size {circumflex over (m)}_(k) forany time t_(k)∈[21, 40]U[1145,1165], t_(k)∈[21, 40]U[2440, 2460],t_(k)∈[21, 40]U[2865, 2885], and t_(k)∈[21, 40]U[375, 395] for naturalgas, crude oil, coal and ethanol data, respectively, is 3≤{circumflexover (m)}_(k)≤20. Further, all comments that are made with regard toTable 2 regarding the four energy commodities remain valid with regardto Table 6.

In Table 7, the real and LLGMM simulated price values of each of thefour energy commodities, including natural gas, crude oil, coal, andethanol, are shown, respectively. The absolute error of each of energycommodity simulated value is also shown.

TABLE 7 Real, Simulation using LLGMM method, and absolute error ofsimulation using starting delay r = 20. Simulated Real y^(s)_({circumflex over (m)}) _(k) _(,k) |Error| t_(k) y_(k) (LLGMM) |y_(k) −y^(s) _({circumflex over (m)}) _(k) _(,k) Natural gas 21 2.759 2.77180.0128 22 2.659 2.6566 0.0024 23 2.742 2.7353 0.0067 24 2.562 2.57570.0137 25 2.495 2.5332 0.0382 26 2.54 2.5336 0.0064 27 2.592 2.56310.0289 28 2.57 2.5797 0.0097 29 2.541 2.4846 0.0564 30 2.618 2.62450.0065 31 2.564 2.5469 0.0171 32 2.667 2.6763 0.0093 33 2.633 2.63080.0022 34 2.515 2.5021 0.0129 35 2.53 2.5136 0.0164 36 2.549 2.54580.0032 37 2.603 2.5835 0.0195 38 2.603 2.5822 0.0208 39 2.603 2.60750.0045 40 2.815 2.8728 0.0578 . . . . . . . . . . . . . . . . . . . . .. . . 1145 5.712 5.7577 0.0457 1146 5.588 5.6488 0.0608 1147 5.6935.7062 0.0132 1148 5.791 5.7917 0.0007 1149 5.614 5.5799 0.0341 11505.442 5.4099 0.0321 1151 5.533 5.5035 0.0295 1152 5.378 5.407 0.029 11535.373 5.3682 0.0048 1154 5.382 5.3827 0.0007 1155 5.507 5.4896 0.01741156 5.552 5.5423 0.0097 1157 5.31 5.318 0.008 1158 5.338 5.3794 0.04141159 5.298 5.3541 0.0561 1160 5.189 5.1838 0.0052 1161 5.082 5.38040.2984 1162 5.082 4.9802 0.1018 1163 5.082 5.1933 0.1113 1164 4.9655.1925 0.2275 1165 4.767 4.7917 0.0247 Crude oil 21 24 24.025 0.025 2223.9 24.093 0.193 23 23.05 23.051 0.001 24 22.3 22.887 0.587 25 22.4522.126 0.324 26 22.35 22.409 0.059 27 21.75 22.12 0.37 28 22.1 22.1370.037 29 22.4 22.315 0.085 30 22.5 22.531 0.031 31 22.65 22.712 0.062 3221.95 22.003 0.053 33 21.6 21.853 0.253 34 21 21.099 0.099 35 20.9521.012 0.062 36 21.1 20.971 0.129 37 20.8 20.786 0.014 38 20.3 20.0480.252 39 20.25 20.244 0.006 40 20.75 20.734 0.016 . . . . . . . . . . .. . . . . . . . . . . . . 2440 57.35 57.376 0.026 2441 56.74 56.4470.293 2442 57.55 57.523 0.027 2443 59.09 58.968 0.122 2444 60.27 60.2780.008 2445 60.75 60.737 0.013 2446 58.41 58.494 0.084 2447 58.72 58.6140.106 2448 58.64 58.95 0.31 2449 57.87 57.865 0.005 2450 59.13 58.9670.163 2451 60.11 59.937 0.173 2452 58.94 59.068 0.128 2453 59.93 60.1410.211 2454 61.18 61.53 0.35 2455 59.66 59.792 0.132 2456 58.59 58.4810.109 2457 58.28 58.224 0.056 2458 58.79 58.928 0.138 2459 56.23 56.3290.099 2460 55.9 54.676 1.224 Coal 21 8.69 8.6747 0.0153 22 8.63 8.61750.0125 23 8.69 8.6862 0.0038 24 8.94 8.9184 0.0216 25 9.31 9.3069 0.003126 8.94 8.8992 0.0408 27 8.94 8.8745 0.0655 28 9.13 9.1162 0.0138 299.19 9.234 0.044 30 8.57 8.5495 0.0205 31 8.69 8.7241 0.0341 32 8.888.8866 0.0066 33 8.57 8.5084 0.0616 34 8.75 8.7447 0.0053 35 8.63 8.60030.0297 36 8.44 8.412 0.028 37 8.44 8.4465 0.0065 38 8.94 8.9538 0.013839 9 9.0064 0.0064 40 8.94 8.8655 0.0745 . . . . . . . . . . . . . . . .. . . . . . . . 2865 29.31 29.291 0.019 2866 28.68 28.8 0.12 2867 26.7726.891 0.121 2868 27.45 27.316 0.134 2869 27 27.189 0.189 2870 26.6726.812 0.142 2871 26.51 26.709 0.199 2872 26.48 26.54 0.06 2873 25.1525.313 0.163 2874 25.57 25.47 0.1 2875 25.88 26.078 0.198 2876 25.2425.208 0.032 2877 25 25.138 0.138 2878 25.08 25.306 0.226 2879 25.0525.16 0.11 2880 25.89 25.509 0.381 2881 25.23 25.278 0.048 2882 25.9425.961 0.021 2883 25.26 25.255 0.005 2884 25.25 25.298 0.048 2885 26.0625.882 0.178 Ethanol 21 1.895 1.9024 0.0074 22 1.95 1.9315 0.0185 231.974 1.9788 0.0048 24 2.7 2.5529 0.1471 25 2.515 2.5134 0.0016 26 2.292.3306 0.0406 27 2.44 2.3718 0.0682 28 2.415 2.3927 0.0223 29 2.3 2.33110.0311 30 2.1 2.072 0.028 31 2.04 2.0323 0.0077 32 2.16 2.1561 0.0039 332.13 2.0796 0.0504 34 2.155 2.2141 0.0591 35 2.01 1.9687 0.0413 36 1.931.8762 0.0538 37 1.9 1.9186 0.0186 38 1.975 1.9052 0.0698 39 1.98 2.0190.039 40 2 1.9385 0.0615 . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . 375 2.073 2.09 0.017 3762.02 2.0589 0.0389 377 2.073 2.0601 0.0129 378 2.065 2.0312 0.0338 3792.055 2.0725 0.0175 380 2.209 2.2254 0.0164 381 2.44 2.462 0.022 3822.517 2.51 0.007 383 2.718 2.6979 0.0201 384 2.541 2.5164 0.0246 3852.566 2.5328 0.0332 386 2.626 2.5831 0.0429 387 2.587 2.5606 0.0264 3882.628 2.6322 0.0042 389 2.587 2.5651 0.0219 390 2.536 2.53 0.006 3912.42 2.4268 0.0068 392 2.247 2.2228 0.0242 393 2.223 2.2072 0.0158 3942.39 2.4141 0.0241 395 2.38 2.4265 0.0465

FIGS. 4A and 4B illustrate real and simulated prices for natural gas andethanol using the local lagged adapted generalized method of momentsdynamic process, respectively, for r=20.

Goodness-of-Ft Measures.

The goodness-of-fit measures are found for four energy commodities,natural gas, crude oil, coal and ethanol. This is achieved by using thefollowing goodness-of-fit measures:

$\begin{matrix}\left\{ \begin{matrix} & = & {\left\lbrack {\frac{1}{N}{\sum\limits_{t = 1}^{N}{\frac{1}{S}{\sum\limits_{s = 1}^{S}\left( {y_{t}^{(s)} - y_{i}} \right)^{2}}}}} \right\rbrack^{\frac{1}{2}},} \\ & = & {{\frac{1}{N}{\sum\limits_{t = 1}^{N}{\underset{s}{median}\left( {{y_{i}^{s} - {\underset{l}{median}\left( y_{i}^{l} \right)}}} \right)}}},} \\ & = & {{\frac{1}{N}{\sum\limits_{i = 1}^{N}\left( {{{\underset{s}{median}\left( y_{i}^{s} \right)} - y_{i}}} \right)}},}\end{matrix} \right. & (69)\end{matrix}$where {y_(t) ^(s)}_(t=1, 2, . . . , N) ^(s=1, 2, . . . , S) is a doublesequence of simulated values at the data collected/observed time t=1, 2,. . . , N, RAMSE is the root mean square error of the simulated path,AMAD is the average median absolute deviation, and AMB is the theaverage median bias. The goodness-of-fit measures are computed usingS=100 pseudo-data sets. The comparison of the goodness-of-fit measuresRAMSE, AMAD, and AMB for the four energy commodities: natural gas, crudeoil, coal, and ethanol data are recorded in Table 8.

Remark 23.

As the RAMSE decreases, then the state estimates approach to the truevalue of the state. As the value of AMAD increases, the influence of therandom environmental fluctuations on the state dynamic processincreases. In addition, if the value of RAMSE decreases and the value ofAMAD increases, then the method of study possesses a greater degree ofability for state and parameter estimation accuracy and greater degreeof ability to measure the variability of random environmentalperturbations on the state dynamic of system. Further, as RAMSEdecreases, AMAD increases, and AMB decreases, the method of studyincreases its performance under the three goodness of fit measures in acoherent way. On other hand, as the RAMSE increases, the state estimatestend to move away from the true value of the state. As the value of AMADdecreases, the influence of the random environmental fluctuations onstate dynamic process decreases.

In addition, if the value of RAMSE increases and the value of AMADdecreases, then the method of study possesses a lesser degree of abilityfor state and parameter estimation accuracy and lesser degree of abilityto measure the variability of random environmental perturbations on thestate dynamic of system. Further, as the RAMSE increases, AMAD decreasesand the AMB increases, the method of study decreases its performanceunder the three goodness-of-fit measures in a coherent manner.

The Comparison of Goodness-of-Fit Measures for r=5, r=10, and r=20.

The following table exhibits the goodness-of-fit measures for the energycommodities natural data, crude oil, coal, and ethanol data using theinitial delays r=5, r=10, and r=20.

TABLE 8 Goodness-of-fit Measures for r = 5, r = 10 and r = 20 Goodnessof-fit Measure Natural gas Crude oil Coal Ethanol r = 5

0.1801 1.1122 1.2235 0.1001 1.1521 24.6476 9.4160 0.3409 1.1372 27.270712.8370 0.3566 r = 10

0.1004 0.5401 0.8879 0.0618 1.1330 24.5376 9.4011 0.3233 1.1371 27.270812.8369 0.3566 r = 20

0.0674 0.4625 0.4794 0.0375 1.1318 24.5010 9.4009 0.3213 1.1374 27.270712.8370 0.3566

Remark 24.

From Tables 3, 5, and 7 it is clear that as r increases the absoluteerror decreases. Furthermore, the comparison of the goodness-of-fitmeasures in Table 8 for the natural gas, crude oil, coal, and ethanoldata the energy commodities using the initial delays r=5, r=10, and r=20shows that as the delay r increases, the root mean square errordecreases significantly, AMAD decreases very slowly, and AMB remainsunchanged.

Remark 25.

Computer readable instructions can be designed to exhibit the flowchartshown in FIGS. 2 and 3. For example, computer readable instructions forparameter estimation, simulations, and forecasting can be written andtested using MATLAB®. Due to the online control nature of m_(k) in ourmodel, it is worth mentioning that the execution times for each of thefour commodities: Natural gas, Crude oil, Coal and Ethanol depend on therobustness of the data.

Illustration 2: Application of Presented Approach to U. S. Treasury BillYield Interest Rate and U.S. Eurocurrency Exchange Rate Data Set.

Here, the conceptual computational algorithm discussed in Section 4 isapplied to estimate the parameters in equation (45) using the real timeU.S. Treasury Bill Yield Interest Rate (U.S. TBYIR) and the U.S.Eurocurrency Exchange Rate (U.S. EER) data collected on Forex database.

Graphical, Simulation and Statistical Results.

Using ϵ=0.001, r=20, and p=2, the ϵ-best sub-optimal estimates ofparameters β, μ, δ, σ and γ for each Treasury bill Yield and U.S.Eurocurrency rate data sets are exhibited in Tables 9 and 10,respectively.

TABLE 9 Estimates for {circumflex over (m)}_(k),

,

,

,

,

 for U.S. Treasury Bill Yield Interest Rate data. interest rate t_(k){circumflex over (m)}_(k)

21 2 1.5199 −7.0332 1.46 0.0446 0.9078 22 2 1.2748 −5.919 1.46 0.09411.5 23 10 2.9904 −13.928 1.46 0.0576 1.5 24 12 1.8604 −8.6515 1.460.0895 1.5 25 6 2.1606 −10.076 1.46 0.1064 1.5 26 20 0.0199 −0.0372 1.460.1097 1.3872 27 16 −0.0274 0.1991 1.46 0.1066 1.4348 28 4 −0.18410.9753 1.46 0.1345 1.2081 29 19 0.3261 −1.3952 1.46 0.1855 0.7006 30 120.2707 −1.1525 1.46 0.1624 1.4187 31 13 0.543 −2.4097 1.46 0.2571 1.498632 11 0.5357 −2.4098 1.46 0.1962 −0.0695 33 11 0.4723 −2.1258 1.460.3494 0.097 34 11 −0.3697 1.4705 1.46 1.4014 1.4983 35 4 −0.7862 3.37031.46 0.3488 1.4993 36 6 −0.3375 1.3041 1.46 0.2914 1.4711 37 5 0.3541−2.0609 1.46 0.2676 1.4972 38 14 0.2368 −1.1239 1.46 0.3201 1.4961 39 81.1109 −5.5453 1.46 0.6811 −0.7462 40 4 1.9032 −9.1187 1.46 1.10550.1008 41 11 0.4364 −2.1327 1.46 0.3532 1.4994 42 4 0.2942 −1.3004 1.460.4885 1.4975 43 5 0.4012 −1.9198 1.46 0.3418 1.5 44 3 0.2605 −1.21081.46 0.4133 1.4705 45 5 0.4213 −2.0086 1.46 0.3324 1.4992 . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .420 12 3.8416 −18.331 1.46 0.1187 1.4906 421 12 2.8918 −13.821 1.460.6961 1.386 422 12 0.5602 −2.6281 1.46 0.3759 1.1741 423 7 0.5825−2.7201 1.46 0.1753 1.4935 424 7 0.7397 −3.4486 1.46 0.1687 0.5396 425 70.2488 −1.1148 1.46 0.1819 0.6161 426 7 0.8447 −3.9535 1.46 0.41820.7124 427 11 −0.2202 1.098 1.46 0.2013 0.6577 428 12 −0.1169 0.62561.46 0.1779 0.6063 429 9 0.1464 −0.6472 1.46 0.3672 1.2589 430 9 0.0343−0.117 1.46 0.3637 0.7374 431 9 0.1785 −0.6832 1.46 0.1395 0.5804 432 19−0.0031 0.1015 1.46 0.1932 1.1832 433 8 0.1651 −0.6463 1.46 0.17450.5374 434 19 0.4102 −1.6622 1.46 0.121 0.3774 435 8 0.2941 −1.1608 1.460.1085 1.0262 436 19 0.3694 −1.4911 1.46 0.1547 1.4945 437 14 1.6473−6.6877 1.46 0.2198 −0.0071 438 5 1.417 −5.7323 1.46 0.1406 −0.1462 43917 1.3024 −5.3352 1.46 0.133 0.2225 440 9 0.2839 −1.191 1.46 0.19290.0883 441 17 0.2053 −0.8785 1.46 0.2007 −0.1338 442 17 −0.4585 1.67541.46 0.4803 0.944 443 7 −0.2917 0.8858 1.46 0.5227 −0.236 444 9 −0.023−0.2999 1.46 0.5836 −0.2083 445 13 −0.3263 1.2217 1.46 0.2632 −0.1684

TABLE 10 Estimates for {circumflex over (m)}_(k),

,

,

,

,

 for U.S. Eurocurrency Exchange Rate. US Eurocurrency Exchange Ratet_(k) {circumflex over (m)}_(k)

21 2 −0.1282 0.1406 1.4892 0.0235 −1.4529 22 3 8.3385 −7.7988 1.48920.0256 1.4954 23 2 3.1279 −2.9205 1.4892 0.0286 1.4995 24 20 0.22−0.1976 1.4892 0.0298 1.4948 25 18 3.0772 −2.8778 1.4892 0.016 1.4741 264 3.8605 −3.6034 1.4892 0.0147 1.3925 27 13 3.7355 −3.4973 1.4892 0.03951.4959 28 16 2.436 −2.2773 1.4892 0.0315 −0.7142 29 17 1.8545 −1.72991.4892 0.0159 −1.4613 30 3 6.4061 −5.9636 1.4892 0.0324 −2.4907 31 121.0648 −0.9689 1.4892 0.0242 1.47 32 15 0.4861 −0.4244 1.4892 0.0285 1.533 18 2.9505 −2.7502 1.4892 0.0267 1.4943 34 5 3.8981 −3.635 1.48920.0984 1.4807 35 4 0.4644 −0.4841 1.4892 0.1052 1.4884 36 3 0.753−0.7159 1.4892 0.0474 1.4954 37 3 0.719 −0.682 1.4892 0.0472 1.4995 38 3−0.7094 0.6544 1.4892 0.0482 1.4948 39 5 1.221 −1.1708 1.4892 0.06491.4741 40 9 6.7537 −6.4315 1.4892 0.0395 1.4959 41 9 1.0019 −0.94391.4892 0.0566 1.4962 42 11 5.5279 −5.2617 1.4892 0.0309 1.499 43 55.3829 −5.1253 1.4892 0.0529 0.1514 44 10 5.2433 −4.9934 1.4892 0.04830.8817 45 10 5.2445 −4.9945 1.4892 0.0305 1.3425 . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 2 10.779−10.219 1.4892 0.0167 0.8188 156 14 2.4641 −2.3297 1.4892 0.0227 0.8437157 4 3.2423 −3.0622 1.4892 0.0184 1.4906 158 6 3.1716 −3.0016 1.48920.0204 0.4736 159 7 6.2013 −5.8656 1.4892 0.0163 0.6027 160 8 9.3459−8.8311 1.4892 0.0207 0.6834 161 4 5.3512 −5.0566 1.4892 0.027 0.4978162 16 −1.3298 1.2689 1.4892 0.0289 0.3431 163 12 4.7287 −4.4662 1.48920.0206 1.2122 164 18 6.22 −5.8772 1.4892 0.0184 1.0666 165 19 13.13−12.394 1.4892 0.021 1.4906 166 18 7.1076 −6.6994 1.4892 0.0211 1.386167 5 3.2762 −3.0824 1.4892 0.0255 1.1741 168 11 3.0507 −2.8403 1.48920.0296 1.4935 169 10 0.9617 −0.8742 1.4892 0.0234 0.5396 170 19 2.0934−1.9275 1.4892 0.027 0.6161 171 5 0.0174 −0.0078 1.4892 0.0275 0.7124172 7 3.2551 −3.0304 1.4892 0.0244 0.6577 173 19 0.909 −0.8452 1.48920.0258 0.6063 174 19 0.8669 −0.807 1.4892 0.0219 1.2589 175 10 1.9332−1.7976 1.4892 0.0189 0.7374 176 10 13.928 −12.966 1.4892 0.0235 0.5804177 6 8.7675 −8.1583 1.4892 0.0232 1.1832 178 9 1.3481 −1.2544 1.48920.0198 0.5374 179 14 0.9565 −0.8852 1.4892 0.0232 0.3774 180 8 0.7656−0.5372 1.4892 0.0132 0.2771

Tables 9 and 10 show the ϵ-best sub-optimal local admissible sample size{circumflex over (m)}_(k) and the corresponding parameter estimatesβ_({circumflex over (m)}) _(k) _(,k), μ_({circumflex over (m)}) _(k)_(,k), δ_({circumflex over (m)}) _(k) _(,k), σ_({circumflex over (m)})_(k) _(,k), and γ_({circumflex over (m)}) _(k) _(,k) for the U. S.Treasury Bill Yield Interest Rate (US-TBYIR) and U. S. EurocurrencyExchange Rate (US-EER) data at each time t_(k), respectively. This isbased on p≤r, and the initial real data time-delay r=20. That is, thedata schedule time t_(r)=t₂₀. Furthermore, note that the range of theϵ-best sub-optimal local admissible sample size for the U. S. TBYIR andU. S. EER data for time t_(k)∈[21,45]∪[420,445] andt_(k)∈[21,45]∪[155,180], respectively, is 2≤{circumflex over(m)}_(k)≤20. All comments made with regard to Table 2 remain valid withregard to Tables 9 and 10 in the context of the the U. S. treasury billYield Interest Rate and the U. S. Eurocurrency Exchange Rate data attime t_(k) and the LLGMM approach.

FIGS. 5A and 5B illustrate real and simulated U.S. treasury billinterest rates and U.S. eurocurrency exchange rates using the locallagged adapted generalized method of moments dynamic process,respectively, with r=20.

Comparison of Goodness-of-Fit Measures for U. S. TBYIR and U. S. EERUsing r=20.

Table 11 compares the Goodness-of-fit Measures for the U. S. TBYIR andU. S. EER data using r=20.

TABLE 11 Goodness-of-fit Measures for the U. S. TBYIR and U. S. EER datausing r = 20. r = 20 Goodness of-fit Meaure U. S. TBYIR U. S. EER

0.0024 0.0137

0.0148 0.0718

0.0165 0.1033

5. Forecasting

Referring back to FIG. 2, at reference numeral 216, the process 200further includes forecasting at least one future state value of thestochastic model of the continuous-time dynamic process using theoptimal m_(k)-local moving sequence. Further, at reference numeral 218,the process 200 includes determining an interval of confidenceassociated with the at least one future state value. In those contexts,the application of the LLGMM approach to robust forecasting and theconfidence interval problems is outlined in this section. It does notrequire a large data size or any type of stationary conditions. First,an outline about forecasting problems is outlined. The ϵ-bestsub-optimal simulated value y_({circumflex over (m)}) _(k) _(,k) ^(s) attime t_(k) is used to define a forecast y_({circumflex over (m)}) _(k)_(,k) ^(f) for y_(k) at the time t_(k) for each of the Energy commoditymodel, and the U. S. TBYIR and U.S. EER.

5.1. Forecasting for Energy Commodity Model

In the context of the illustration in Section 3.5, we begin forecastingfrom time t_(k). Using the data up to time t_(k−1), we compute{circumflex over (m)}_(i), σ_({circumflex over (m)}) _(i) _(,i) ²,a_({circumflex over (m)}) _(i) _(,i) and μ_({circumflex over (m)}) _(i)_(,i) for i∈I₀(k−1). We assume that we have no information about thereal data {y_(i)}_(i=k) ^(N). Under these considerations, imitating thecomputational procedure outlined in Section 4 and using equation (43),we find the estimate of the forecast y_({circumflex over (m)}) _(k)_(,k) ^(f) at time t_(k) by employing the following discrete timeiterative process:y _({circumflex over (m)}) _(k) _(,k) ^(f) =y _({circumflex over (m)})_(k−1) _(,k−1) ^(s) +a _({circumflex over (m)}) _(k−1) _(,k−1) y_({circumflex over (m)}) _(k−1) _(,k−1) ^(s)(μ_({circumflex over (m)})_(k−1) _(,k−1) −y _({circumflex over (m)}) _(k−1) _(,k−1) ^(s))Δt+σ_({circumflex over (m)}) _(k−1) _(,k−1) y _({circumflex over (m)})_(k−1) _(,k−1) ^(s) ΔW _(k),  (70)where the estimates σ_({circumflex over (m)}) _(k−1) _(,k−1) ²,a_({circumflex over (m)}) _(k−1) _(,k−1) and μ_({circumflex over (m)})_(k−1) _(,k−1) are defined in (43) with respect to the known past dataup to the time t_(k−1). We note that y_({circumflex over (m)}) _(k)_(,k) ^(f) is the ϵ-sub-optimal estimate for y_(k) at time t_(k).

To determine y_({circumflex over (m)}) _(k+1) _(,k+1) ^(f), we needσ_({circumflex over (m)}) _(k) _(,k) ², a_({circumflex over (m)}) _(k)_(,k) and μ_({circumflex over (m)}) _(k) _(,k). Since we only haveinformation of real data up to time t_(k−1), we use the forecastedestimate y_({circumflex over (m)}) _(k) _(,k) ^(f) as the estimate ofy_(k) at time t_(k), and to estimate σ_({circumflex over (m)}) _(k)_(,k) ², a_({circumflex over (m)}) _(k) _(,k) andμ_({circumflex over (m)}) _(k) _(,k). Hence, we can writea_({circumflex over (m)}) _(k) _(,k) as

a_(m̂_(k), k) ≡ a_(m̂_(k), y_(k − m̂_(k) + 1), y_(k − m̂_(k) + 2, …, y_(k − 1), y_(m_(k), k)^(f))).

We can also re-write

μ_(m̂_(k), k) ≡ μ_(m̂_(k), y_(k − m̂_(k) + 1), y_(k − m̂_(k) + 2, …, y_(k − 1), y_(m_(k), k)^(f))).To find y_({circumflex over (m)}) _(k+2) _(,k−2) ^(f), we use theestimates

a_(m̂_(k + 1), k + 1) ≡ a_(m̂_(k + 1), y_(k − m̂_(k) + 2), y_(k − m̂_(k) + 3, …, y_(k − 1), y)_( _(m̂_(k), k)^(f), y_(m̂_(k + 1), k + 1)^(f)))and

μ_(m̂_(k + 1), k + 1) ≡ μ_(m̂_(k + 1), y_(k − m̂_(k) + 2), y_(k − m̂_(k) + 3, …, y_(k − 1), y_(m̂_(k), k)^(f)y_(m̂_(k + 1), k + 1)^(f))).Continuing this process in this manner, we use the estimates

a_(m̂_(k + i − 1), k + i − 1) ≡ a_(m̂_(k + i − 1), y_(k − m̂_(k) + i), y_(k − m̂_(k) + i + 1), …, y_(k − 1), y_(m̂_(k), k)^(f)y_(m̂_(k + 1), k + 1)^(f), …, y_(m̂_(k + 1), k + i − 1)^(f))and

μ_(m̂_(k + i − 1), k + i − 1) ≡ μ_(m̂_(k + i − 1), y_(k − m̂_(k) + i), y_(k − m̂_(k) + i + 1), …, y_(k − 1), y_(m̂_(k), k)^(f), y_(m̂_(k + 1), k + 1)^(f), …, y_(m̂_(k + 1), k + i − 1)^(f))to estimate y_({circumflex over (m)}) _(k+i) _(,k+i) ^(f).

5.1.1. Prediction/Confidence Interval for Energy Commodities

In order to be able to assess the future certainty, we also discussabout the prediction/confidence interval. We define the 100(1−α)%confidence interval for the forecast of the statey_({circumflex over (m)}) _(i) _(,i) ^(f) at time t_(i), i≥k, asy_({circumflex over (m)}) _(i) _(,i)^(f)±z_(1−α/2)(s_({circumflex over (m)}) _(i−1) _(,i−1) ²)^(1/2)y_({circumflex over (m)}) _(i−1) _(,i−1) ^(f) where(s_({circumflex over (m)}) _(i−1) _(,i−1) ²)^(1/2)y_({circumflex over (m)}) _(i−1) _(,i−1) ^(f) is the estimate for thesample standard deviation for the forecasted state derived from thefollowing iterative processy _({circumflex over (m)}) _(k) _(,k) ^(f) =y _({circumflex over (m)})_(k−1) _(,k−1) ^(f) +a _({circumflex over (m)}) _(k−1) _(,k−1) y_({circumflex over (m)}) _(k−1) _(,k−1) ^(f)(μ_({circumflex over (m)})_(k−1) _(,k−1) −y _({circumflex over (m)}) _(k−1) _(,k−1) ^(f))Δt+σ_({circumflex over (m)}) _(k−1) _(,k−1) y _({circumflex over (m)})_(k−1) _(,k−1) ^(f) ΔW _(k).  (71)

It is clear that the 95% confidence interval for the forecast at timet_(i) is

(y_(m̂_(i), i)^(f) − 1.96(s_(m̂_(i − 1), i − 1)²)^(1/2)y_(m̂_(i − 1), i − 1)^(f), y_(m̂_(i), i)^(f) + 1.96(s_(m̂_(i − 1), i − 1)²)^(1/2)y_(m̂_(i − 1), i − 1)^(f)),where the lower end denotes the lower bound of the state estimate andthe upper end denotes the upper bound of the state estimate.

FIGS. 6A and 6B show the graphs of the forecast and 95 percentconfidence limit for the daily Henry Hub Natural gas and weekly Ethanoldata, respectively. Further, 6A and 6B show two regions: the simulationregion S and the forecast region F. For the simulation region S, we plotthe real data together with the simulated data. For the forecast regionF, we plot the estimate of the forecast as explained in Section 5. Theupper and the lower simulated sketches in FIGS. 6A and 6B arecorresponding to the upper and lower ends of the 95% confidenceinterval. Next, we show graphs which exhibit the bounds of the estimatesof the forecast for the four energy commodity.

5.2. Prediction/Confidence Interval for U. S. Treasury Bill YieldInterest Rate and U. S. Eurocurrency Rate

Following the same procedure explained in Section 5.1, we show the graphof the real, simulated, forecast and 95% confidence limit for the U. S.TBYIR and U.S. EER for the initial delay r=20. FIG. 7A shows the real,simulated, forecast, and 95 percent confidence limit for the Interestrate data, and FIG. 7B shows the real, simulated, forecast, and 95%confidence level for the U. S. EER.

6. The Byproduct of the Llgmm Approach

The DTIDMLSMVSP not only plays role (a) to initiate ideas for the usageof discrete time interconnected dynamic approach parallel to thecontinuous-time dynamic process, (b) to speed-up the computation time,and (c) to significantly reduce the state error estimates, but it alsoprovides an alternative approach to the GARCH(1,1) model and comparableresults with ex post volatility results of Chan et al. Furthermore, theLLGMM directly generates a GMM based method (e.g., Remark 12, Section3). In this section, we briefly discuss these comparisons in the contextof four energy commodity and U.S. TBYIR and EER data.

6.1 Comparison Between DTIDMLSMVSP and GARCH Model

In this subsection, we briefly compare the applications of DTIDMLSMVSPand GARCH in the context of four energy commodities. In reference toRemark 6, we compare the estimates s_({circumflex over (m)}) _(k) _(,k)² with the estimate derived from the usage of a GARCH(1,1) modeldescribed defined by

$\begin{matrix}{\left. z_{t} \middle| {\sim {\left( {0,h_{t}} \right)}} \right.,{h_{t} = {\alpha_{0} + {\alpha_{1}h_{t - 1}} + {\beta_{1}z_{t - 1}^{2}}}},{\alpha_{0} > 0},\alpha_{1},{\beta_{1} \geq 0.}} & (72)\end{matrix}$The parameters α₀, α₁, and β₁ of the GARCH(1,1) conditional variancemodel (72) for the four commodities natural gas, crude oil, coal, andethanol are estimated. The estimates of the parameters are given inTable 12.

TABLE 12 Parameter estimates for Garch(1,1) Model (72). Data Set α₀ α₁β₁ Natural Gas 6.863 × 10⁻⁵ 0.853 0.112 Crude Oil 9.622 × 10⁻⁵ 0.9170.069 Coal 3.023 × 10⁻⁵ 0.903 0.081 Ethanol 4.152 × 10⁻⁴ 0.815 0.019

We later show a side by side comparison of s_({circumflex over (m)})_(k) _(,k) ² and the volatility described by GARCH(1,1) model describedin (72) with coefficients in Table 12. The GARCH model does not estimatevolatility but instead demonstrated insensitivity.

6.2 Comparison of DTIDMLSMVSP with Chan et al

In this subsection, using the U.S. TBYIR and U.S. EER data, thecomparison between the DTIDMLSMVSP and ex post volatility of Chan et alis made. According to the work of Chan et al, we define the ex postvolatility by the absolute value of the change in U.S. TBYIR data.Likewise, we define simulated volatility by the square root of theconditional variance implied by the estimates of the model (45). Using(45), we calculate our simulated volatility as

σ_(m̂_(k), k)(y_(m̂_(k), k)^(s))^(δ_(m̂_(k), k))).We compare our work (DTIDMLSMVSP) with FIG. 1 of Chan et al. Their modeldoes not clearly estimate the volatility. It demonstrated insensitivityin the sense that it was unable to capture most of the spikes in theinterest rate ex post volatility data.

6.3 Formulation of Aggregated Generalized Method of Moment (AGMM)

In this subsection, using the theoretical basis of the LLGMM and Remark12 (Section 3), we develop a GMM based method for state and parameterestimation problems.

6.3.1. AGMM Method Applied to Energy Commodities

Using the aggregated parameter estimates ā, μ, and σ² described by themean value of the estimated samples {a_({circumflex over (m)}) _(i)_(,i)}_(i=0) ^(N), {μ_({circumflex over (m)}) _(i) _(,i)}_(i=0) ^(N) and{σ_({circumflex over (m)}) _(i) _(,i) ²}_(i=0) ^(N), respectively, wediscuss the simulated price values for the four energy commodities. Wedefine

${\overset{\_}{a} = {\frac{1}{N}{\sum\limits_{i = 0}^{N}a_{{\hat{m}}_{i},i}}}},{\overset{\_}{\mu} = {\frac{1}{N}{\sum\limits_{i = 0}^{N}\mu_{{\hat{m}}_{i},i}}}},{{{and}\mspace{14mu}\overset{\_}{\sigma^{2}}} = {\frac{1}{N}{\sum\limits_{i = 0}^{N}\sigma_{{\hat{m}}_{i},i}^{2}}}},$respectively. Further, ā, μ, and σ² are referred to as aggregatedparameter estimates of a, μ, and σ² over the given entire finiteinterval of time, respectively.

These estimates are derived using the following discretized system:

$\begin{matrix}{{y_{i}^{ag} = {y_{i - 1}^{ag} + {{\overset{\_}{a}\left( {\overset{\_}{\mu} - y_{i - 1}^{ag}} \right)}y_{i - 1}^{ag}\Delta\; t} + {{\overset{\_}{\sigma^{2}}}^{1/2}y_{i - 1}^{ag}\Delta\; W_{i}}}},} & (73)\end{matrix}$where y_(k) ^(ag) denotes the simulated value for y_(k) at time t_(k) attime. The overall descriptive data statistic regarding the four energycommodity prices and estimated parameters are recorded in the Table 13.

TABLE 13 Descriptive statistics for a, μ and σ² with time delay r = 20.Data Set Y Y Std (Y) Δln (Y) var (Δln(Y)) ā Std (a) μ Std (μ) σ² std(σ²) 95% C.I. μ Nat. Gas  4.5504  1.5090 0.0008 0.0015 0.1867 0.3013 4.5538  2.3565 0.0013 0.0017 (4.4196, 4.6880) Crude Oil 54.0093 31.02480.0003 0.0006 0.0215 0.0517 54.0307 37.4455 0.0005 0.0008 (51.8978,56.1636) Coal 27.1441 17.8394 0.0003 0.0015 0.0464 0.0879 27.056721.3506 0.0014 0.0022 (25.8405, 28.2729) Ethanol  2.1391  0.4455 0.00110.0020 0.3167 0.8745  2.1666  0.7972 0.0018 0.0030 (2.0919, 2.2414)

Table 13 shows the descriptive statistics for a, μ and σ² with timedelay r=20. Further, μ is approximately close to the overall descriptivestatistics of the mean Y of the real data for each of the energycommodities shown in column 2. Also, σ² is approximately close to theoverall descriptive statistics of the variance of Δln(Y)=ln(Y_(i))−ln(Y_(i−1)) in column 5. Further, column 12 shows thatthe mean of the actual data set in Column 2 falls within the 95%confidence interval of μ. This exhibits that the parameterμ_({circumflex over (m)}) _(k) _(,k) is the mean level of y_(k) at timet_(k).

Using the aggregated parameter estimates ā, μ, and σ² in Table 13(columns 6, 8, and 10), the simulated price values for the four energycommodities are shown in columns 3, 6, 9 and 12 of Table 14.

TABLE 14 Real, Simulation using AGMM with r = 20. Natural gas Crude oilCoal Ethanol Simulated Simulated Simulated Simulated y_(k) ^(ag) y_(k)^(ag) y_(k) ^(ag) y_(k) ^(ag) t_(k) Real (AGMM) t_(k) Real (AGMM) t_(k)Real (AGMM) t_(k) Real (AGMM) 21 2.759 2.649 21 24.00 23.974 21 8.6909.111 21 1.895 1.834 22 2.659 2.651 22 23.900 24.204 22 8.630 9.028 221.950 1.854 23 2.742 2.636 23 23.050 25.229 23 8.690 9.192 23 1.9741.798 24 2.562 2.625 24 22.300 25.586 24 8.940 9.032 24 2.700 1.858 252.495 2.593 25 22.450 26.470 25 9.310 8.938 25 2.515 1.830 26 2.54 2.52526 22.350 25.953 26 8.940 8.792 26 2.290 1.954 27 2.592 2.513 27 21.75026.229 27 8.940 9.035 27 2.440 1.926 28 2.57 2.399 28 22.100 26.555 289.130 9.255 28 2.415 1.939 29 2.541 2.485 29 22.400 26.402 29 9.1909.018 29 2.300 1.883 30 2.618 2.506 30 22.500 27.34 30 8.570 8.687 302.100 1.880 31 2.564 2.460 31 22.650 26.24 31 8.690 8.985 31 2.040 1.81732 2.667 2.295 32 21.950 26.765 32 8.880 9.339 32 2.160 1.810 33 2.6332.534 33 21.600 26.358 33 8.570 9.359 33 2.130 1.774 34 2.515 2.514 3421.000 26.87 34 8.750 9.310 34 2.155 1.717 35 2.53 2.573 35 20.95026.835 35 8.630 9.302 35 2.010 1.658 36 2.549 2.592 36 21.100 26.725 368.440 9.543 36 1.930 1.607 37 2.603 2.456 37 20.800 26.439 37 8.4409.288 37 1.900 1.645 38 2.603 2.428 38 20.300 26.916 38 8.940 9.155 381.975 1.635 39 2.603 2.505 39 20.250 26.989 39 9.000 8.469 39 1.9801.629 40 2.815 2.526 40 20.750 26.759 40 8.940 8.899 40 2.00 1.745 . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11455.712 5.218 2440 57.350 48.179 2865 29.310 17.839 375 2.073 2.625 11465.588 5.414 2441 56.740 48.239 2866 28.680 18.563 376 2.02 2.784 11475.693 5.460 2442 57.550 46.984 2867 26.770 19.577 377 2.073 2.558 11485.791 5.464 2443 59.090 47.418 2868 27.450 19.841 378 2.065 2.670 11495.614 5.544 2444 60.270 48.137 2869 27.000 18.876 379 2.055 2.565 11505.442 5.700 2445 60.750 49.185 2870 26.670 18.465 380 2.209 2.796 11515.533 5.710 2446 58.410 48.271 2871 26.510 18.139 381 2.44 2.783 11525.378 5.936 2447 58.720 48.384 2872 26.480 17.963 382 2.517 2.659 11535.373 5.869 2448 58.640 47.509 2873 25.150 18.151 383 2.718 2.739 11545.382 5.778 2449 57.870 48.654 2874 25.570 17.987 384 2.541 2.681 11555.507 5.732 2450 59.130 46.883 2875 25.880 18.393 385 2.566 2.631 11565.552 5.816 2451 60.110 46.403 2876 25.240 18.492 386 2.626 2.638 11575.31 6.000 2452 58.940 45.564 2877 25.000 18.621 387 2.587 2.542 11585.338 6.162 2453 59.930 44.177 2878 25.080 18.806 388 2.628 2.491 11595.298 5.899 2454 61.180 43.112 2879 25.050 19.384 389 2.587 2.392 11605.189 6.008 2455 59.660 43.47 2880 25.890 20.131 390 2.536 2.393 11615.082 6.175 2456 58.590 41.531 2881 25.230 21.099 391 2.42 2.534 11625.082 6.191 2457 58.280 40.452 2882 25.940 21.499 392 2.247 2.687 11635.082 5.814 2458 58.790 41.968 2883 25.260 21.38 393 2.223 2.701 11644.965 5.701 2459 56.230 44.359 2884 25.250 20.786 394 2.39 2.703 11654.767 5.871 2460 55.90 44.679 2885 26.060 20.892 395 2.38 2.655 11664.675 5.998 2461 56.420 43.081 2886 26.030 21.269 396 2.366 2.559 11674.79 5.952 2462 58.010 44.235 2887 26.660 20.371 397 2.335 2.575 11684.631 5.782 2463 57.280 43.199 2888 27.120 19.822 398 2.428 2.466 11694.658 5.673 2464 60.30 42.655 2889 26.400 19.644 399 2.409 2.369 11704.57 5.936 2465 60.970 43.498 2890 26.940 20.602 400 2.29 2.222

TABLE 15 Comparison of Goodness-of-fit Measures for the LLGMM and AGMMmethod using initial delay r = 20. Goodness LLGMM AGMM of-fit NaturalCrude Natural Crude Measure Gas Oil Coal Ethanol gas oil Coal Ethanol

0.0674 0.4625 0.4794 0.0375 1.4968 30.7760 17.7620 0.4356

1.1318 24.5010 9.4009 0.3213 0.0068 0.0857 0.0833 0.0035

1.1371 27.2707 12.8370 0.3566 1.2267 27.3050 13.1060 0.3579

6.3.2. Formulation of Aggregated Generalized Method of Moment (AGMM) forU.S. Treasury Bill and U.S. Eurocurrency Rate

The overall descriptive statistics of data sets regarding U. S. TreasuryBill Yield Interest Rate and U. S. Eurocurrency Exchange Rate arerecorded in the following table for initial delay r=20.

TABLE 17 Y Std (Y) β Std (β) μ Std (μ) δ Sts (δ) σ Std (σ) γ Std (γ)Descriptive statistics for β, μ, δ, σ, and γ for the U.S. TBYIR datawith initial delay r = 20 0.05667 0.0268 0.8739 1.8129 −3.8555 8.76081.4600 0.00 0.3753 0.5197 1.4877 0.1357 Descriptive statistics for β, μ,δ, σ, and γ for the U.S. Eurocurrency Exchange Rate data with initialdelay r = 20 1.6249 0.1337 1.5120 2.1259 −1.1973 1.6811 1.4892 0.000.0243 0.0180 1.08476 1.0050

In Tables 16 and 17, the real and the LLGMM simulated rates of the U.S.TBYIR and the U. S. Eurocurrency Exchange Rate (US-EER) are exhibited inthe first and second columns, respectively. Using the aggregatedparameter estimates β, μ, δ, σ and {circumflex over (γ)} in therespective Tables 16 (columns: 3, 5, 7, 9 and 11) and Table 17 (columns:3, 5, 7, 9, and 11), the simulated rates for the U.S. TYBIR and the U.S.EER are shown in the column 3 of Table 18. These estimates are derivedusing the following discretized system:y _(i) ^(ag) =y _(i−1) ^(ag)+(β y _(i−1) ^(ag)+μ(y _(i−1) ^(ag)) ^(δ))+σ(y _(i−1) ^(af)) ^(γ) ΔW _(i),  (74)where AGMM, y_(k) ^(ag), y_(k) at time t_(k) are defined in (73).

TABLE 18 Estimates for Real, Simulated value using LLGMM and AGMMmethods for U.S. TYBIR and the U.S. EER, respectively for initial delayr = 20. Interest Rate Data Eurocurrency Rate Simulated Simulated RealSimulated Simulated t_(k) Real LLGMM AGMM t_(k) Real LLGMM AGMM 210.0465 0.0459 0.0326 21 1.7448 1.6732 1.655 22 0.0459 0.0467 0.0299 221.7465 1.7711 1.6588 23 0.0462 0.0463 0.0342 23 1.7638 1.7588 1.6096 240.0464 0.0463 0.034 24 1.874 1.8423 1.6251 25 0.045 0.0457 0.0365 251.7902 1.7971 1.6221 26 0.048 0.048 0.0447 26 1.7635 1.7668 1.5984 270.0496 0.0496 0.0449 27 1.74 1.7362 1.6368 28 0.0537 0.053 0.0538 281.7763 1.7755 1.5795 29 0.0535 0.0529 0.0535 29 1.8219 1.8224 1.5708 300.0532 0.0536 0.0489 30 1.8985 1.9002 1.6174 31 0.0496 0.0495 0.0575 311.9166 1.8897 1.6403 32 0.047 0.0479 0.0548 32 1.992 1.9361 1.6425 330.0456 0.0453 0.0385 33 1.7741 1.7738 1.6409 34 0.0426 0.0423 0.042 341.5579 1.5601 1.6759 35 0.0384 0.0413 0.0339 35 1.5138 1.5017 1.5287 360.036 0.0363 0.0384 36 1.5102 1.5028 1.5445 37 0.0354 0.0358 0.0457 371.4832 1.5171 1.6334 38 0.0421 0.0434 0.0321 38 1.4276 1.4353 1.6666 390.0427 0.043 0.023 39 1.51 1.4972 1.606 40 0.0442 0.044 0.0299 40 1.57341.588 1.662 41 0.0456 0.0463 0.0301 41 1.5633 1.5556 1.6305 42 0.04730.0462 0.0365 42 1.4966 1.4856 1.5987 43 0.0497 0.0512 0.0341 43 1.48681.4914 1.5832 44 0.05 0.0505 0.042 44 1.4864 1.4785 1.621 45 0.04980.0497 0.0451 45 1.4965 1.4854 1.6208 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . 420 0.045 0.04490.0337 155 1.5581 1.5635 1.6326 421 0.0457 0.045 0.0309 156 1.60971.6195 1.574 422 0.0455 0.0459 0.0389 157 1.6435 1.6089 1.6232 4230.0472 0.047 0.0306 158 1.5793 1.5817 1.6669 424 0.0468 0.0464 0.0385159 1.5782 1.5826 1.649 425 0.0486 0.0481 0.0179 160 1.6108 1.62061.5725 426 0.0507 0.0499 0.0191 161 1.6368 1.6256 1.6879 427 0.0520.0514 0.0257 162 1.662 1.644 1.6681 428 0.0532 0.0539 0.029 163 1.61151.6156 1.6534 429 0.0555 0.0546 0.0379 164 1.571 1.5708 1.6387 4300.0569 0.0588 0.0404 165 1.6692 1.6912 1.6243 431 0.0566 0.056 0.0487166 1.6766 1.6832 1.5822 432 0.0579 0.0587 0.0432 167 1.7188 1.72241.5764 433 0.0569 0.0571 0.0436 168 1.7856 1.7285 1.6206 434 0.05960.0602 0.0393 169 1.8225 1.7952 1.6044 435 0.0609 0.0601 0.04 170 1.86991.8896 1.6792 436 0.06 0.0601 0.0483 171 1.8562 1.8964 1.5417 437 0.06110.0604 0.0292 172 1.772 1.7717 1.6087 438 0.0617 0.0617 0.031 173 1.83981.8372 1.5426 439 0.0577 0.0583 0.0379 174 1.8207 1.8214 1.6147 4400.0515 0.0509 0.0464 175 1.8248 1.8242 1.6544 441 0.0488 0.05 0.0476 1761.7934 1.7795 1.5929 442 0.0442 0.0441 0.0516 177 1.7982 1.8056 1.5845443 0.0387 0.0445 0.0675 178 1.8335 1.835 1.6625 444 0.0362 0.03130.0484 179 1.934 1.9301 1.5832 445 0.0349 0.0386 0.0484 180 1.90541.8939 1.5472

In Table 18, we show a side by side comparison of the estimates for thesimulated value using LLGMM and AGMM methods for U.S. Treasury BillYield Interest Rate and U.S. Eurocurrency Exchange Rate, respectively:initial delay r=20.

7. Comparisons of LLGMM with OCBGMM

In this section, we briefly compare LLGMM and OCBGMM in the frame-workof the conceptual, computational, mathematical, and statistical resultscoupled with role, scope and applications. For this purpose, to betterappreciate and understand the comparative work, we utilize the state andparameter estimation problems for the stochastic dynamic model ofinterest rate that has been studied extensively in the frame-work oforthogonality condition vector based generalized method of moments(OCBGMM). Recall that the LLGMM approach is based on seven interactivecomponents (Section 1). On the other hand, the existing OCBGMM (GMM andIRGMM) approach and its extensions are based on five components (Section3). The basis for the formation of orthogonality condition parametervectors (OCPV) in the LLGMM (Section 3) and OCBGMM (GMM/IRGMM) aredifferent. In the existing OCBGMM (GMM/IRGMM), the orthogonalitycondition vectors are formed on the basis of algebraic manipulationcoupled with econometric specification-based discretization scheme(OCPV-Algebraic) rather than stochastic calculus and a continuous-timestochastic dynamic model based OCPV-Analytic. This motivates to extend acouple of OCBGMM-based state and parameter estimation methods.

Using the stochastic calculus based formation of the OCPV-Analytic inthe context of the continuous-time stochastic dynamic model (Section 3),two new OCBGMM based methods are developed for the state and parameterestimation problems. The proposed OCBGMM methods are direct extensionsof the existing OCBGMM method and its extension IRGMM in the context ofthe OCPV. In view of this difference and for the sake of comparison, thenewly developed OCBGMM and the existing OCBGMM methods are referred toas the OCBGMM-Analytic and OCBGMM-Algebraic, respectively. Inparticular, the GMM and IRGMM with OCPV-algebraic are denoted asGMM-Algebraic and IRGMM-Algebraic and corresponding extensions under theOCPV-Analytic as GMM-Analytic and IRGMM-Analytic, respectively.

Furthermore, using LLGMM based method, the aggregated generalized methodof moments (AGMM) introduced in Subsection 3.5 and described inSubsection 6.3 is also compared along with the above stated methods,namely GMM-Algebraic, GMM-Analytic, IRGMM-Algebraic, and IRGMM-Analytic.A comparative analysis of the results of GMM-Algebraic, GMM-Analytic,IRGMM-Algebraic, IRGMM-Analytic and AGMM methods with the LLGMM for thestate and parameter estimation problems of the interest rate and energycommodities stochastic dynamic models are briefly outlined in thesubsequent subsections. First, based on Sections 1, 2, 3 and 4, webriefly summarize the comparison between the LLGMM and OCBGMM methods.

7.1 Theoretical Comparison Between LLGMM and OCBGMM

Based on the foundations of the analytical, conceptual, computational,mathematical, practical, statistical, and theoretical motivations anddevelopments outlined in Sections 2, 3, 4 and 5, we summarize thecomparison between the innovative approach LLGMM with the existing andnewly developed OCBGMM methods in separate tables in a systematicmanner.

Table 19 outlines the differences between the LLGMM method and existingorthogonality condition based GMM/IRGMM-Algebraic and the newlyformulated GMM/IRGMM-Analytic methods together with the AGMM.

TABLE 19 Mathematical Comparison Between the LLGMM and OCBGMM OCBGMM-OCGMM- Feature LLGMM Algebraic Analytic Justifications Composition:Seven components Five components Five components Sections 1. 3 Model:Development Selection Development/Selection Sections 1, 3 Goal:Validation Specification/Testing Validation/Testing Sections 1, 3Discrete-Time Constructed Using Econometric Constructed Remarks 8,15Scheme: from SDE specification from SDE Formation of Using stochasticFormed using Using Stochastic Remarks 7, 8, Orthogonality calculusalgebraic calculus 9, 14, 15 Vector: manipulation

TABLE 20 Intercomponent Interaction Comparison Between LLGMM and OCBGMMOCBGMM- OCGMM- Feasture LLGMM Algebraic Analytic Justifications MomentEquations: Local Lagged Single/global Single/global Remarks 5, 8, 18a,adaptive process system system and 18b Type of Moment Local laggedSingle-shot Single-shot Remarks 5, 8, 13, Equations: adaptive process15, and 16 Component Strongly Weakly Weakly Remarks 8, 13, 14,Interconnections: connected connected connected 15, 16, and 18 DynamicDiscrete-time Static Static Remarks 5, 8, 18 and and Static: DynamicLemma 1 (Section 2)

TABLE 21 Conceptual Computational Comparison Between LLGMM and OCBGMMOCBGMM- OCGMM- Feature LLGMM Algebraic Analytic Justifications Localadmissible Multi- Single- Singe- Definition 10, Remark 18, Lagged Datachoice choice/data choice/data Subsection 4.2 Size: size size Localadmissible Multi- Single- Single- Adapted finite restricted class oflagged choice choice/data choice/data sample data: Definition 11, finiterestriction sequence sequence Remark 18, Subsection 4.2 sequences Localadmissible Multi- Single-shot Single-shot Subsection 4.2 finite sequencechoice estimate estimates parameter estimates: Local admissible Multi-Single- Single- Remark 18, Subsection 4.3 sequence of choice choicechoice finite state simulation values: Quadratic Mean Multi- Single-Single- Remark 18, Subsection 4.3 Square ϵ-sub- choice error erroroptimal errors: ϵ-sub-optimal Multi- Single- Single- Definition 12,Remark 18, local lagged choice choice choice Subsection 4.3 sample size:ϵ-best sub optimal ϵ-best sub No-choice No-choice Remark 18, Subsection4.3 sample size: optimal choice ϵ-best sub optimal ϵ-best No-choiceNo-choice Remark 18, Subsection 4.3 parameter estimated: estimatorsϵ-best sub optimal ϵ-best No-choice No-choice Remark 18, Subsection 4.3state estimate sub optimal choice

TABLE 22 Theoretical Performance Comparison Between LLGMM and OCBGMMOCBGMM- OCGMM- Feature LLGMM Algebraic Analytic Justifications DataSize: Reasonable Size Large Data Size Large Data For Respectable Sizeresults Stationary Condition: Not required Need Ergotic/ Need ForReasonable Asympotic Ergodic/ results stationary Asymptotic Multi-levelAt least 2 level Single-shot Single-shot Not comparable optimization:hierarchical optimization Admissible Strategies: Multi-choicesSingle-shot Single-shot Not comparable Computational Stability:Algorithm Single-choice Single-choice Simulation Converges in a resultssingle/double digit trials Significance of lagged Stabilizing agentNon-existence of the Non- Not comparable adaptive process: featureexistence Operation: Operates like Operates like a static Operates likeObvious, details Discrete time dynamic process static process seeSections 4, Dynamic Process 5, 6 and 7

7.2 Comparisons of LLGMM Method with Existing Methods Using InterestRate Stochastic Model

The continuous-time interest rate process is described by a nonlinearItô-Doob-type stochastic differential equation:dy=(α+βy)dt+σy ^(γ) dW(t).  (75)

The energy commodities stochastic dynamic model is described in (27), inSubsection 3.5. These models would be utilized to further compare therole, scope and merit of the LLGMM and OCBGMM methods in the frame-workof the graphical, computational and statistical results and applicationsto forecasting and prediction with certain degree of confidence.

Remark 26.

The continuous-time interest rate model (75) was chosen so that we cancompare our LLGMM method with the OCBGMM method. Our proposed model forthe continuous-time interest rate model is described in (45). We willlater compare the results derived using model (75) with the resultsusing (45) from Subsections 3.6 and 4.6.

Descriptive Statistic for Time-Series Data Set.

For this purpose, first, we consider one month risk free rates from theMonthly Interest rate data sets for the period Jun. 30, 1964 to Dec. 31,2004. Table 23 below shows some statistics of the data set shown in FIG.19.

TABLE 23 Statistics for the Interest Rate data for Jun. 30, 1964 to Dec.31, 2004. Variable N Mean Std dev ρ₁ ρ₂ ρ₃ ρ₄ ρ₅ ρ₆ y_(t) 487 0.05920.0276 0.9809 0.9508 0.9234 0.8994 0.8764 0.8519 Δy_(t) 486 −0.000030.0050 0.3305 −0.0919 −0.1048 −0.0351 0.0403 −0.1877

Mean, standard deviations, and autocorrelations of monthly Treasury billyields (US TBYIR) and yield changes ρ_(j) denotes the autocorrelationcoefficient of order j, N represents the total number of observationsused.

The Orthogonality Condition Vector for (75).

First, we present the orthogonality condition parameter vectors (OCPV)for the GMM-Algebraic, GMM-Analytic, IRGMM-Algebraic, and IRGMM-Analyticmethods. These orthogonality vectors are then used for the state andparameter estimation problems. For this, we need to follow the procedure(Section 3) for obtaining the analytic orthogonality condition parametervector (OCPV-Analytic). We consider the Lyapunov functions (

${V_{1}\left( {t,y} \right)} = {\frac{1}{2}y^{2}}$and

${V_{2}\left( {t,y} \right)} = {\frac{1}{3}{y^{3}.}}$The Itô-differential of V₁ and V₂ with respect to (75) are:

$\begin{matrix}\left\{ {\begin{matrix}{{d\left( {\frac{1}{2}y^{2}} \right)} = {{\left\lbrack {{a\; y} + {\beta\; y^{2}} + {\frac{1}{2}\sigma^{2}y^{2\;\gamma}}} \right\rbrack{dt}} + {\sigma\; y^{\gamma + 1}{{dW}(t)}}}} \\{{\left. {{d\left( {\frac{1}{3}y^{3}} \right)} = {\left\lbrack {{\alpha\; y^{2}} + {\beta\; y^{3}}} \right) + {\sigma^{2}y^{{2\;\gamma} + 1}}}} \right\rbrack{dt}} + {\sigma\; y^{\gamma + 2}d\;{W(t)}}}\end{matrix}.} \right. & (76)\end{matrix}$

The component of orthogonality condition vector (OCPV-Analytic) isdescribed by:

$\begin{matrix}\left\{ \begin{matrix}{{\Delta\; y_{t}} - \left( {{\left\lbrack y_{t} \middle| \mathcal{F}_{t - 1} \right\rbrack} - y_{t - 1}} \right)} \\{{\frac{1}{2}\Delta\;\left( y_{t}^{2} \right)} - {\frac{1}{2}\left( {{\left\lbrack y_{t}^{2} \middle| \mathcal{F}_{t - 1} \right\rbrack} - y_{t - 1}^{2}} \right)}} \\{{\frac{1}{3}\Delta\;\left( y_{t}^{3} \right)} - {\frac{1}{3}\left( {{\left\lbrack y_{t}^{3} \middle| \mathcal{F}_{t - 1} \right\rbrack} - y_{t - 1}^{3}} \right)}} \\{{\left\lbrack \left( {{\Delta\; y_{t}} - {\left\lbrack {\Delta\; y_{t}} \middle| \mathcal{F}_{t - 1} \right\rbrack}} \right)^{2} \middle| \mathcal{F}_{t - 1} \right\rbrack} - {\sigma^{2}y_{t - 1}^{2\gamma}\Delta\;{t.}}}\end{matrix} \right. & (77)\end{matrix}$where

$\begin{matrix}\left\{ {\begin{matrix}{\left\lbrack \;\left. y_{t} \middle| \mathcal{F}_{t - 1} \right. \right\rbrack - y_{t - 1}} & = & {\left( {\alpha + {\beta\; y}} \right)\Delta\; t} \\{\frac{1}{2}\left( {{\left\lbrack y_{t}^{2} \middle| \mathcal{F}_{t - 1} \right\rbrack} - y_{t - 1}^{2}} \right)} & = & {\left\lbrack {{\alpha\; y_{t - 1}} + {\beta\; y_{t - 1}^{2}} + {\frac{1}{2}\sigma^{2}y_{t - 1}^{2\;\gamma}}} \right\rbrack\Delta\; t} \\{\frac{1}{3}\left( {{\left\lbrack y_{t}^{3} \middle| \mathcal{F}_{t - 1} \right\rbrack} - y_{t - 1}^{3}} \right)} & = & {\left\lbrack {{\alpha\; y_{t - 1}^{2}} + {\beta\; y_{t - 1}^{3}} + {\sigma^{2}y_{t - 1}^{{2\;\gamma} + 1}}} \right\rbrack\Delta\; t} \\{\left\lbrack \left( {{\Delta\; y_{t}} - {\left\lbrack {\Delta\; y_{t}} \middle| \mathcal{F}_{t - 1} \right\rbrack}} \right)^{2} \middle| \mathcal{F}_{t - 1} \right\rbrack} & = & {\sigma^{2}y_{t - 1}^{2\gamma}\Delta\; t}\end{matrix}.} \right. & (78)\end{matrix}$

On the other hand, using discrete time econometric specification coupledwith algebraic manipulations, the components of orthogonality conditionparameter vector (OCPV-Algebraic) are as follows:

$\begin{matrix}\left\{ \begin{matrix}{y_{t} - y_{t - 1} - {\left( {\alpha + {\beta\; y}} \right)\Delta\; t}} \\{y_{t - 1}\left( {y_{t} - y_{t - 1} - {\left( {\alpha + {\beta\; y}} \right)\Delta\; t}} \right)} \\{\left( {y_{t} - y_{t - 1} - {\left( {\alpha + {\beta\; y}} \right)\Delta\; t}} \right)^{2} - {\sigma^{2}y_{t - 1}^{2\gamma}}} \\{y_{t - 1}\left\lbrack {\left( {y_{t} - y_{t - 1} - {\left( {\alpha + {\beta\; y}} \right)\Delta\; t}} \right)^{2} - {\sigma^{2}y_{t - 1}^{2\;\gamma}}} \right\rbrack}\end{matrix} \right. & (79)\end{matrix}$

We apply the GMM-Algebraic, IRGMM-Algebraic, GMM-Analytic, andIRGMM-Analytic methods.

Parameter Estimates of (75) Using LLGMM Method.

Using the LLGMM method, the parameter estimatesα_({circumflex over (m)}) _(k) _(,k), β_({circumflex over (m)}) _(k)_(,k), σ_({circumflex over (m)}) _(k) _(,k), andγ_({circumflex over (m)}) _(k) _(,k) are shown in Table 24. Here, we useϵ=0.001, p=2, and initial delay r=20.

TABLE 24 Estimates for {circumflex over (m)}_(k),α_({circumflex over (m)}) _(k) _(,k), β_({circumflex over (m)}) _(k)_(,k), σ_({circumflex over (m)}) _(k) _(,k), γ_({circumflex over (m)})_(k) _(,k) for U. S. Treasury Bill Yield Interest Rate data using LLGMM.Interest Rate t_(k) {circumflex over (m)}_(k) α_({circumflex over (m)})_(k) _(,k) β_({circumflex over (m)}) _(k) _(,k)σ_({circumflex over (m)}) _(k) _(,k) γ_({circumflex over (m)}) _(k)_(,k) 21 2 0.0334 −0.7143 0.0446 1.5 22 3 0.0427 −0.9254 0.0766 1.5 23 40.0425 −0.9198 0.0914 1.5 24 5 0.0413 −0.8937 0.09 1.5 25 4 0.1042−2.2619 0.1003 1.5 26 19 0.0002 0.0083 0.1043 1.5 27 14 0.0024 −0.03590.1281 1.5 28 5 −0.023 0.5207 0.3501 1.5 29 13 0.0037 −0.0573 0.1652 1.530 18 0.0008 0.001 0.1447 1.5 31 3 −0.3827 7.1316 0.26 1.5 32 19 0.006−0.1213 0.1828 1.5 33 6 0.0063 −0.1359 0.343 1.5 34 19 0.0081 −0.17050.1993 1.5 35 4 −0.0166 0.2984 0.3509 1.5 36 4 −0.0059 0.0721 0.2318 1.537 9 −0.0035 0.0324 0.3114 1.5 38 14 0.0051 −0.1186 0.3385 1.5 39 200.0059 −0.1294 0.282 1.5 40 12 0.0075 −0.185 0.3447 1.5 41 12 0.0099−0.2379 0.3579 1.5 42 4 −0.0089 0.2335 0.3562 1.5 43 7 0.0074 −0.12890.4654 1.5 44 7 0.0182 −0.3677 0.4206 1.5 45 6 0.0106 −0.2031 0.2356 1.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .420 3 0.0836 −1.9 0.1006 1.5 421 8 0.0428 −0.9671 0.783 1.5 422 3 0.0359−0.7857 0.1702 1.5 423 8 0.0127 −0.2766 0.1719 1.5 424 6 0.0178 −0.38570.1636 1.5 425 6 0.0177 −0.3685 0.1829 1.5 426 18 0.0146 −0.3172 0.38711.5 427 8 0.0017 −0.012 0.1788 1.5 428 4 0.009 −0.1489 0.1341 1.5 429 9−0.0059 0.1469 0.1616 1.5 430 13 −0.0046 0.116 0.191 1.5 431 9 0.0039−0.0532 0.1369 1.5 432 9 0.0027 −0.0287 0.1109 1.5 433 3 0.0857 −1.50.0952 1.5 434 9 0.0102 −0.1661 0.1197 1.5 435 9 0.0075 −0.114 0.107 1.5436 5 0.029 −0.485 0.1446 1.5 437 4 0.0476 −0.784 0.2163 1.5 438 90.0122 −0.1966 0.1054 1.5 439 4 0.1626 −2.6824 0.1248 1.5 440 20 0.0072−0.1278 0.1916 1.5 441 19 0.0084 −0.1502 0.2016 1.5 442 17 0.0024−0.0479 0.2369 1.5 443 7 −0.0153 0.2236 0.2687 1.5 444 3 0.0054 −0.21880.3887 1.5 445 16 −0.0076 0.1177 0.2528 1.5

Table 24 shows the parameter estimates of {circumflex over (m)}_(k),α_({circumflex over (m)}) _(k) _(,k), β_({circumflex over (m)}) _(k)_(,k), σ_({circumflex over (m)}) _(k) _(,k), γ_({circumflex over (m)})_(k) _(,k) in the model (75) for U.S. Treasury Bill Yield Interest Ratedata. As noted before, the range of the ϵ-best sub-optimal localadmissible sample size {circumflex over (m)}_(k) for any timet_(k)∈[21,45]U[420,445] is 2≤{circumflex over (m)}_(k)≤20. We also drawthe similar conclusions (a) to (e) as outlined in Remark 20.

Parameter Estimates of (75) Using OCBGMM Methods.

Following Remark 12, we define the average α, β, σ, and γ by

$\begin{matrix}\left\{ \begin{matrix}{{\alpha = {\frac{1}{N}{\sum\limits_{k = 1}^{N}\alpha_{{\overset{.}{m}}_{k},k}}}},} \\{\overset{\_}{\beta} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}\beta_{{\overset{.}{m}}_{k},k}}}} \\{{\overset{\_}{\sigma} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}\sigma_{{\overset{.}{m}}_{k},k}}}},} \\{{\overset{\_}{\gamma} = {\frac{1}{N}{\sum\limits_{k = 1}^{N}\gamma_{{\overset{.}{m}}_{k},k}}}},}\end{matrix} \right. & (80)\end{matrix}$where the parameters α_({circumflex over (m)}) _(k) _(,k),β_({circumflex over (m)}) _(k) _(,k), σ_({circumflex over (m)}) _(k)_(,k), γ_({circumflex over (m)}) _(k) _(,k) are each estimated in Table25 at time t_(k) using LLGMM method.

Imitating the argument used in Subsection 6.3, the parameters and stateare also estimated. These parameter estimates are shown in the row ofAGMM approach in Table 25. We also estimate the parameters in (75) byfollowing both the GMM-algebraic and GMM-analytic frame-work. Similarly,the parameter estimates (75) are determined under the IRGMM-algebraicand IRGMM-analytic approaches. These parameter estimates are recorded inrows of GMM-algebraic, GMM-analytic, IRGMM-algebraic, and IRGMM-analyticapproaches, respectively, in Table 25.

Comparison of Goodness-of-Fit Measures.

In order to statistically compare the different estimation techniques weestimate the statistics RAMSE, AMAD, and AMB defined in (69). Thegoodness-of-fit measures are computed using S=100 pseudo-data sets ofthe same sample size, and the real data set, N=487 months. Thet-statistics of each parameter estimate is in parenthesis, the smallestvalue of RAMSE for all method is italicized. The goodness-of-fitmeasures RAMSE, AMAD and AMB are recorded under the columns 6, 7, and 8respectively.

TABLE 25 Comparison of parameter estimates of model (75) and thegoodness-of- fit measures RAMSE, AMAD, and AMB under the usage of GMM-Algebraic, GMM-Analytic, IRGMM-Algebraic, IRGMM-Analytic, AGMM, andLLGMM methods. Method α β σ γ

GMM- 0.0017 −0.0308 0.4032 1.5309 0.0424  0.0098 0.0195 Algebraic (1.53)(−1.33) (1.55) (3.21) GMM- 0.0009 −0.0153 0.0184 0.4981 0.0315  0.01610.0190 Analytic (1.06) (−0.90) (1.25) (1.73) IRGMM- 0.0020 −0.0410 0.2071.3031 0.03186 0.00843 0.01972 Algebraic (0.32) (−0.21) (0.25) (1.02)IRGMM- 0.0084 −0.1436 0.1075 1.3592 0.0278  0.0028 0.01968 Analytic(0.44) (−0.40) (0.22) (1.01) AGMM 0.0084 −0.1436 0.1075 1.3592 0.0288 0.0047 0.0207 (0.41) (−0.33) (0.25) (0.98) LLGMM 0.0027* 0.0146 0.0178

The LLGMM estimates are derived using initial delay r=20, p=2 andϵ=0.001. Among these stated methods, the LLGMM method generates thesmallest RAMSE value. In fact, the RAMSE value is smaller than the onetenth of any other RAMSE values. Further, second, third and fourthsmaller RAMSE values are due to the IRGMM-Analytic, AGMM andGMM-Analytic methods, respectively. This exhibits the superiority of theLLGMM method over all other methods. We further observe that the LLGMMapproach yields the smallest AMB in comparison with the OCBGMMapproaches. The GMM-Analytic, IRGMM-Analytic and IRGMM-Algebraic rankthe second, third and fourth smaller values, respectively. The highvalue of AMAD for the LLGMM method signifies that the LLGMM captures theinfluence of random environmental fluctuations on the dynamic ofinterest rate process. We further note that the first, second, third,and fourth smaller AMB values are due to the GMM-Analytic, LLGMM,IRGMM-Algebraic, and GMM-Algebraic methods, respectively. Again, fromRemark 23, the smallest RAMSE, higher AMAD, and smallest AMB value underthe LLGMM method exhibit the superior performance under the threegoodness-of-fit measures. We also notice that the performance ofstochastic calculus based-OCPV-Analytic methods, namely, GMM-Analytic,IRGMM-Analytic and AGMM is better than the performance of OCPV-Algebraicbased, GMM-Algebraic, and IRGMM-Algebraic approaches. In short, thissuggests that the OCPV-Analytic based GMM methods are more superior thanthe OCPV-Algebraic based GMM methods.

TABLE 26 Parameter estimates and goodness of fit tests for one monthrisk free rates for periods June 1964-December 1981 and January1982-December 2004. June 1964- January 1982- Orthogonality December 1981December 2004 Condition

GMM-Algebraic 0.0468 0.0377 GMM-Analytic 0.0315 0.0347 IRGMM-Algebraic0.0307 0.0326 IRGMM-Analytic 0.0200 0.0215 LLCIMM 0.0030* 0.0017*

Table 26 shows the goodness-of-fit measures RAMSE using GMM-Algebraic,GMM-Analytic, IRGMM-Algebraic, IRGMM-Analytic, and LLGMM method for twoseparate sub-periods: 06/1964-12/1981 and 01/1982-12/2004. Among allmethods, the LLGMM method generates the smallest RAMSE value for eachsub-period. Further, the goodness-of-fit measure RAMSE regarding theLLGMM method is less than the one sixth, and one twelfth of any otherRAMSE value, respectively. The IRGMM-Analytic, IRGMM-Algebraic,GMM-Analytic, and GMM-Algebraic methods are in second, third, fourth andfifth place.

Comparative Analysis of Forecasting with 95% Confidence Intervals.

Using data set June 1964 to December 1989, the parameters of model (75)are estimated. Using these parameter estimates, we forecasted themonthly interest rate for Jan. 1, 1990 to Dec. 31, 2004.

TABLE 27 Parameter estimates in (75) in the context of the data fromJune 1964 to December 1989. Method α β σ γ GMM-Algebraic 0.0033 −0.0510.4121 1.5311 GMM-Analytic 0.0009 −0.0155 0.0197 0.4854 IRGMM-Algebraic0.0023 −0.0421 0.3230 1.3112 IRGMM-Analytic 0.0084 −0.1436 0.1073 1.3641AGMM 0.01.54 −0.2497 0.2949 1.4414

7.3 Comparisons of LLGMM Method with Existing and Newly IntroducedOCBGMM Methods Using Energy Commodity Stochastic Model

Using the stochastic dynamic model in (27) of energy commodityrepresented by the stochastic differential equationdy=ay(μ−y)dt+σ(t,y _(t))ydW(t),y(t ₀)=y ₀,  (81)the orthogonality condition parameter vector (OCPV) is described in (30)in Remark 9.

Based on a discretized scheme using the econometric specification, theorthogonality condition parameter vector in the context of algebraicmanipulation is as:

$\begin{matrix}\left\{ {\begin{matrix}{y_{t} - y_{t - 1} - {{{ay}_{t - 1}\left( {\mu - y_{t - 1}} \right)}\Delta\; t}} \\{y_{t - 1}\left( {y_{t} - y_{t - 1} - {{{ay}_{t - 1}\left( {\mu - y_{t - 1}} \right)}\Delta\; t}} \right)} \\\left. {\left( {y_{t} - y_{t - 1} - {{{ay}_{t - 1}\left( {\mu - y_{t - 1}} \right)}\Delta\; t}} \right)^{2} - {\sigma^{2}y_{t - 1}^{2}}} \right\rbrack\end{matrix}.} \right. & (82)\end{matrix}$The goodness-of-fit measures are computed using pseudo-data sets of thesame sample size as the real data set: (i) N=1184 days for natural gasdata, (ii) N=4165 days for crude oil data, (iii) N=3470 for coal data,and (iv) N=438 weeks for ethanol data. The smallest value of RAMSE forall method is italicized.

TABLE 28 Parameter estimates of model (75) and the goodness-of-fitmeasures RAMSE, AMAD, and AMB using GMM-Algebraic, GMM-Analytic,IRGMM-Algebraic, IRGMM-Analytic, AGMM and LLGMM methods for natural gasdata Method a μ σ²

GMM- 0.0023 5.3312 0.0019 1.5119 0.0663 1.1488 Algebraic GMM- 0.00185.4106 0.0015 1.5014 0.0538 1.1677 Analytic IRGMM- 0.2000 4.4996 0.00101.4985 0.0050 1.2299 Algebraic IRGMM- 0.1998 4.4917 0.0011 1.4901 0.00441.2329 Analytic AGMM 0.1867 4.5538 0.0013 1.4968 0.0068 1.2267 LLGMM 0.0674* 1.1318 1.1371

TABLE 29 Parameter estimates of model (75) and the goodness-of-fitmeasures RAMSE, AMAD, and AMB using GMM-Algebraic, GMM-Analytic,IRGMM-Algebraic, IRGMM-Analytic, AGMM and LLGMM methods for crude oildata Method a μ σ²

GMM- 0.0023 54.4847 0.0005 39.2853 0.3577 29.1587 Algebraic GMM- 0.002151.2145 0.0006 38.8007 0.5181 28.7414 Analytic IRGMM- 0.0000 88.59510.0005 30.7511 0.0920 27.5791 Algebraic IRGMM- 0.0021 51.2195 0.000528.9172 0.2496 27.3564 Analytic AGMM 0.0215 54.0307 0.0005 30.776 0.0857 27.3050 LLGMM  0.4625* 24.501  27.2707

TABLE 30 Parameter estimates of model (75) and the goodness-of-fitmeasures RAMSE, AMAD, and AMB using GMM-Algebraic, GMM-Analytic,IRGMM-Algebraic, IRGMM-Analytic, AGMM and LLGMM methods for coal dataMethod a μ σ²

GMM- 0.0000 94.4847 0.0006 22.6866 0.2015 16.3444 Algebraic GMM- 0.000094.4446 0.0006 21.6564 0.2121 16.3264 Analytic IRGMM- 0.0027 34.48380.0013 17.6894 0.3438 13.4981 Algebraic IRGMM- 0.0021 23.1151 0.000517.6869 0.3448 13.4989 Analytic AGMM 0.0464 27.0567 0.0014 17.76200.0833 13.106  LLGMM  0.4794* 9.4009 12.8370

TABLE 31 Parameter estimates of model (75) and the goodness-of-fitmeasures RAMSE, AMAD, and AMB using GMM-Algebraic, GMM-Analytic,IRGMM-Algebraic, IRGMM-Analytic, AGMM and LLGMM methods for ethanolMethod a μ σ²

GMM- 0.0000 94.4847 0.0006 22.6866 0.2015 16.3444 Algebraic GMM- 0.000094.4446 0.0006 21.6564 0.2121 16.3264 Analytic IRGMM- 0.0014  3.45060.0026  0.5844 0.0322  0.4346 Algebraic IRGMM- 0.0015  3.4506 0.0026 0.5813 0.0336  0.4303 Analytic AGMM 0.3167  2.166 0.0018  0.4356 0.0035 0.3579 LLGMM  0.0375* 0.3213  0.3566

Tables 28, 29, 30, and 31 show a comparison parameter estimates of model(75) and the goodness-of-fit measures RAMSE, AMAD, and AMB usingGMM-Algebraic, GMM-Analytic, IRGMM-Algebraic, IRGMM-Analytic, AGMM andLLGMM methods for the daily natural gas data, daily crude oil data,daily coal data, and weekly ethanol data, respectively. The LLGMMestimates are derived using initial delay r=20, p=2 and ϵ=0.001. Amongall methods under study, the LLGMM method generates the smallest RAMSEvalue. In fact, the RAMSE value is smaller than the 1/22, 1/62, 1/36,and 1/10 of any other RAMSE values regarding the natural gas, crude oil,coal and ethanol, respectively. This exhibits the superiority of theLLGMM method over all other methods. We further observe that the LLGMMapproach yields the smallest AMB and highest AMAD value regarding thenatural gas, crude oil, coal and ethanol. The high value of AMAD for theLLGMM method signifies that the LLGMM captures the influence of randomenvironmental fluctuations on the dynamic of energy commodity process.From Remark 23, the smallest RAMSE, highest AMAD, and smallest AMB valueunder the LLGMM method exhibit the superior performance under the threegoodness-of-fit measures.

Ranking of Methods Under Goodness of Fit Measure.

TABLE 32 Ranking of natural gas, crude oil, coal, and ethanol underthree statistical measures RANK OF METHODS UNDER GOODNESS OF FIT MEASURENatural gas Crude oil Coal Ethanol Method

GMM-Algebraic 6 2 2 6 3 6 6 5 6 6 3 6 GMM-Analytic 5 3 3 5 2 5 5 4 5 5 25 IRGMM-Algebraic 4 5 5 3 5 4 4 3 3 4 5 4 IRGMM-Analytic 2 6 6 2 4 3 3 24 3 4 3 AGMM 3 4 4 4 6 2 2 6 2 2 6 2 LLGMM 1 1 1 1 1 1 1 1 1 1 1 1

Remark 27.

The ranking of LLGMM is top one in all three goodness-of-fit statisticalmeasures for all four energy commodity data sets. Further, one of theIRGMM-Analytic and AGMM is ranked either as top 2nd or 3rd under RAMSEmeasure. This exhibits the influence of the usage of stochastic calculusbased orthogonality condition parameter vectors (OCPV-Analytic).

7.4 Comparison of Goodness of Fit Measures of Model (45) with (75) UsingLLGMM Method

As stated in Remark 26, we compare the Goodness of fit Measures RAMSE,AMAD, and AMB using the U.S. Treasury Bill Interest Rate data and theLLGMM applied to the model validation problems of two proposedcontinuous-time dynamic models of U.S. Treasury Bill Interest Rateprocess described by (45) and (75). The LLGMM state estimates of (45)and (75) are computed under the same initial delay r=20, p=2, andϵ=0.001. The results are recorded in the following table.

TABLE 33 Comparison of goodness of fit measure of model (45) with model(75) LLGMM

Model (45) 0.0024* 0.0145 0.0178 Model (75) 0.0027  0.0146 (10178

Table 33 shows that the goodness-of-fit measures RAMSE, AMAD, and AMB ofthe LLGMM method using both models (75) and (45) are very close. Model(45) appears to have the least RAMSE value. This shows that the LLGMMresult performs better using model (45) than using model (75) since ithas a lower root mean square error. The AMAD value using (75) is largerthan the value using (45). This suggests that the influence of therandom environmental fluctuations on state dynamic model (75) is higherthan using the model (45). The AMB value derived using both modelsappeared to be the same, indicating that both model give the sameaverage median bias estimates. Based on this statistical analysis, weconclude that (45) is most appropriate continuous-time stochasticdynamic model for the short-term riskless rate model which includes manywell-known interest rate models.

8. Comparison of LLGMM with Existing Nonparametric Statistical Methods

In this section, we compare our LLGMM method with existing nonparametricmethods. We consider the following existing nonparametric methods.

8.1 Nonparametric Estimation of Nonlinear Dynamics by Metric-Based LocalLinear Approximation (LLA)

The LLA method assumes no functional form of a given model but estimatesfrom experimental data by approximating the curve implied by thefunction by the tangent plane around the neighborhood of a tangentpoint. Suppose the state of interest x_(t) at time t is differentiablewith respect to t and satisfies dx_(t)=f(x_(t))dt, where f:

→

is a smooth map, x_(t)∈

. The approximation of the curve f(x_(t)) in a neighbourhoodU_(ϵ)(x₀)={x: d(x,x₀)<ϵ} is defined by a tangent plane at x₀

${y_{r} = {{f\left( x_{0} \right)} + {\sum\limits_{i = 1}^{k}{\frac{\partial f}{\partial x_{i}}\left( x_{0} \right)\left( {x_{i} - x_{0}} \right)}}}},$where d is a metric on

^(k). Allowing error in the equation and assigning a weight w(x_(t)) toeach error terms ϵ_(t), the method reduces to estimating parameters

${\beta_{i} = {\frac{\partial f}{\partial x_{i}}\left( x_{0} \right)}},{1 = 1},2,\ldots\mspace{14mu},k$in the equation

${{w(t)}y_{t}} = {{\beta_{0} \cdot {w\left( x_{t} \right)}} + {\sum\limits_{i = 1}^{k}{{\beta_{i} \cdot {w\left( x_{t} \right)}}{\left( {x_{t,i} - x_{0,i}} \right).}}}}$

Applying the standard linear regression approach, the least squareestimate {circumflex over (β)} is given by{circumflex over (β)}=({tilde over (X)} ^(T) {tilde over (X)})⁻¹ {tildeover (X)} ^(T) {tilde over (Y)},  (83)where

$\begin{matrix}{{{\overset{\sim}{x}}_{i} = \left( {{{w\left( x_{t_{1}} \right)}\left( {x_{t_{1},1} - x_{0,i}} \right)},\ldots\mspace{14mu},{{w\left( x_{t_{n}} \right)}\left( {x_{t_{n},i} - x_{0,i}} \right)}} \right)^{T}},{i = 1},\ldots\mspace{14mu},k} \\{\overset{\sim}{w} = \left( {{w\left( x_{{t\;}_{1}} \right)},\ldots\mspace{14mu},{w\left( x_{t_{n}} \right)}} \right)^{T}} \\{\overset{\sim}{Y} = \left( {{{w\left( x_{t_{1}} \right)}y_{t_{1}}},\ldots\mspace{14mu},{{w\left( x_{t_{n}} \right)}y_{t_{n}}}} \right)^{T}} \\{\overset{\sim}{X} = {\left( {\overset{\sim}{w},{\overset{\sim}{x}}_{1},\ldots\mspace{14mu},{\overset{\sim}{x}}_{k}} \right).}}\end{matrix}.$

Particularly, the trajectory f(x_(t) _(i) ) is estimated by choosingx₀=x_(t) _(i) , for each i=1, 2, . . . , n, respectively. We use d(x,x₀)=|x−x₀|, where |.| is the standard Euclidean metric on

^(k), and w(x)=ϕ(d(x, x₀)), where ϕ(u)=K(u/ϵ) and K is the EpanechnikovKernel K(x)=0.75(1−x²)₊.

8.2 Risk Estimation and Adaptation after Coordinate Transformation(REACT) Method

Given n pairs of observations (x₁, Y₁), . . . , (x_(n), Y_(n)), theREACT method, the response variable Y is related to the covariate x(called a feature) by the equationY _(i) =r(x _(i))+σϵ_(i),  (84)where ϵ_(i)˜N(0, 1) are IID, and x_(i)=i/n, i=1, 2, . . . , n. Thefunction r(x) is approximated using orthogonal cosine basis ϕ_(i), i=1,2, 3, . . . of [0,1] described byϕ₁(x)≡1, ϕ_(j)(x)=√{square root over (2)}cos((j−1)πx),j≥2.  (85)

The function r(x), expanded as

$\begin{matrix}{{r(x)} = {\sum\limits_{j = 1}^{\infty}{\theta_{j}{\phi_{j}(x)}}}} & (86)\end{matrix}$where

θ_(j) = ∫₀¹ϕ_(j)(x)r(x) dxis approximated. The function estimator

${\hat{r}(x)} = {\sum\limits_{j = 1}^{\hat{J}}{Z_{j}{\phi_{j}(x)}}}$where

${Z_{j} = {\frac{1}{n}{\sum\limits_{i = 1}^{n}{Y_{i}{\phi_{j}\left( x_{i} \right)}}}}},{j = 1},2,\ldots\mspace{11mu},n$and Ĵ is found so that the risk estimator

${\hat{R}(J)} = {\frac{J\;{\hat{\sigma}}^{2}}{n} + {\sum\limits_{j = {J + 1}}^{n}\left( {Z_{j}^{2} - \frac{{\hat{\sigma}}^{2}}{n}} \right)}}$is minimized, {circumflex over (σ)}² is the estimator of variance ofZ_(j).

8.3 Exponential Moving Average Method (EMA)

The EMA for an observation y_(t) at time t may be calculated recursivelyasS _(t) =αy _(t)+(1−α)S _(t−1) , t=1,2,3, . . . ,n,  (87)where 0<α≤1 is a constant that determines the depth of memory of S_(t).

8.4 Goodness-of-Fit Measures for the LLA, REACT, and EMA Methods

In this subsection, we show the goodness-of-fit measures for the LLA,REACT, and EMA methods. We use Ĵ=183 for the REACT method and α=0.5 forthe EMA method.

TABLE 34 Goodness-of-fit measures for the LLA, REACT, and EMA methods.Goodness of-fit Measure Natural gas Crude oil Coal Ethanol LLGMM method

0.0674 0.4625 0.4794 0.0375

1.1318 24.5010 9.4009 0.3213

1.1371 27.2707 12.8370 0.3566 LLA Method

0.3114 1.9163 2.1645 0.2082

1.1406 24.3266 9.4511 0.3290

1.2375 27.2713 12.8388 0.3677 REACT method

0.1895 2.0377 2.0162 0.0775

1.1779 24.6967 9.3791 0.3291

1.12352 27.2711 12.8369 0.3566 EMA method

0.1222 0.7845 0.8233 0.0682

1.1336 24.5858 9.4183 0.3159

1.2352 27.2710 12.8370 0.3567

Comparison of the results derived using these non-parametric methodswith the LLGMM method show that the results derived using the LLGMMmethod is far better than results of the nonparametric methods.

FIG. 8 illustrates an example schematic block diagram of the computingdevice 100 shown in FIG. 1 according to various embodiments describedherein. The computing device 100 includes at least one processingsystem, for example, having a processor 802 and a memory 804, both ofwhich are electrically and communicatively coupled to a local interface806. The local interface 806 can be embodied as a data bus with anaccompanying address/control bus or other addressing, control, and/orcommand lines.

In various embodiments, the memory 804 stores data and software orexecutable-code components executable by the processor 802. For example,the memory 804 can store executable-code components associated with thevisualization engine 130 for execution by the processor 802. The memory804 can also store data such as that stored in the device data store120, among other data.

It is noted that the memory 804 can store other executable-codecomponents for execution by the processor 802. For example, an operatingsystem can be stored in the memory 804 for execution by the processor802. Where any component discussed herein is implemented in the form ofsoftware, any one of a number of programming languages can be employedsuch as, for example, C, C++, C #, Objective C, JAVA®, JAVASCRIPT®,Perl, PHP, VISUAL BASIC®, PYTHON®, RUBY, FLASH®, or other programminglanguages.

As discussed above, in various embodiments, the memory 804 storessoftware for execution by the processor 802. In this respect, the terms“executable” or “for execution” refer to software forms that canultimately be run or executed by the processor 802, whether in source,object, machine, or other form. Examples of executable programs include,for example, a compiled program that can be translated into a machinecode format and loaded into a random access portion of the memory 804and executed by the processor 802, source code that can be expressed inan object code format and loaded into a random access portion of thememory 804 and executed by the processor 802, or source code that can beinterpreted by another executable program to generate instructions in arandom access portion of the memory 804 and executed by the processor802, etc.

An executable program can be stored in any portion or component of thememory 804 including, for example, a random access memory (RAM),read-only memory (ROM), magnetic or other hard disk drive, solid-state,semiconductor, or similar drive, universal serial bus (USB) flash drive,memory card, optical disc (e.g., compact disc (CD) or digital versatiledisc (DVD)), floppy disk, magnetic tape, or other memory component.

In various embodiments, the memory 804 can include both volatile andnonvolatile memory and data storage components. Volatile components arethose that do not retain data values upon loss of power. Nonvolatilecomponents are those that retain data upon a loss of power. Thus, thememory 804 can include, for example, a RAM, ROM, magnetic or other harddisk drive, solid-state, semiconductor, or similar drive, USB flashdrive, memory card accessed via a memory card reader, floppy diskaccessed via an associated floppy disk drive, optical disc accessed viaan optical disc drive, magnetic tape accessed via an appropriate tapedrive, and/or other memory component, or any combination thereof. Inaddition, the RAM can include, for example, a static random accessmemory (SRAM), dynamic random access memory (DRAM), or magnetic randomaccess memory (MRAM), and/or other similar memory device. The ROM caninclude, for example, a programmable read-only memory (PROM), erasableprogrammable read-only memory (EPROM), electrically erasableprogrammable read-only memory (EEPROM), or other similar memory device.

The processor 802 can be embodied as one or more processors 802 and thememory 804 can be embodied as one or more memories 804 that operate inparallel, respectively, or in combination. Thus, the local interface 806facilitates communication between any two of the multiple processors802, between any processor 802 and any of the memories 804, or betweenany two of the memories 804, etc. The local interface 806 can includeadditional systems designed to coordinate this communication, including,for example, a load balancer that performs load balancing.

As discussed above, the LLGMM dynamic process module 130 can beembodied, at least in part, by software or executable-code componentsfor execution by general purpose hardware. Alternatively the same can beembodied in dedicated hardware or a combination of software, general,specific, and/or dedicated purpose hardware. If embodied in suchhardware, each can be implemented as a circuit or state machine, forexample, that employs any one of or a combination of a number oftechnologies. These technologies can include, but are not limited to,discrete logic circuits having logic gates for implementing variouslogic functions upon an application of one or more data signals,application specific integrated circuits (ASICs) having appropriatelogic gates, field-programmable gate arrays (FPGAs), or othercomponents, etc.

The flowchart or process diagrams in FIGS. 2 and 3 are representative ofcertain processes, functionality, and operations of the embodimentsdiscussed herein. Each block can represent one or a combination of stepsor executions in a process. Alternatively or additionally, each blockcan represent a module, segment, or portion of code that includesprogram instructions to implement the specified logical function(s). Theprogram instructions can be embodied in the form of source code thatincludes human-readable statements written in a programming language ormachine code that includes numerical instructions recognizable by asuitable execution system such as the processor 802. The machine codecan be converted from the source code, etc. Further, each block canrepresent, or be connected with, a circuit or a number of interconnectedcircuits to implement a certain logical function or process step.

Although the flowchart or process diagrams in FIGS. 2 and 3 illustrate aspecific order, it is understood that the order can differ from thatwhich is depicted. For example, an order of execution of two or moreblocks can be scrambled relative to the order shown. Also, two or moreblocks shown in succession in FIGS. 2 and 3 can be executed concurrentlyor with partial concurrence. Further, in some embodiments, one or moreof the blocks shown in FIGS. 2 and 3 can be skipped or omitted. Inaddition, any number of counters, state variables, warning semaphores,or messages might be added to the logical flow described herein, forpurposes of enhanced utility, accounting, performance measurement, orproviding troubleshooting aids, etc. It is understood that all suchvariations are within the scope of the present disclosure.

Also, any logic or application described herein, including the LLGMMdynamic process module 130 that are embodied, at least in part, bysoftware or executable-code components, can be embodied or stored in anytangible or non-transitory computer-readable medium or device forexecution by an instruction execution system such as a general purposeprocessor. In this sense, the logic can be embodied as, for example,software or executable-code components that can be fetched from thecomputer-readable medium and executed by the instruction executionsystem. Thus, the instruction execution system can be directed byexecution of the instructions to perform certain processes such as thoseillustrated in FIGS. 2 and 3. In the context of the present disclosure,a “non-transitory computer-readable medium” can be any tangible mediumthat can contain, store, or maintain any logic, application, software,or executable-code component described herein for use by or inconnection with an instruction execution system.

The computer-readable medium can include any physical media such as, forexample, magnetic, optical, or semiconductor media. More specificexamples of suitable computer-readable media include, but are notlimited to, magnetic tapes, magnetic floppy diskettes, magnetic harddrives, memory cards, solid-state drives, USB flash drives, or opticaldiscs. Also, the computer-readable medium can include a RAM including,for example, an SRAM, DRAM, or MRAM. In addition, the computer-readablemedium can include a ROM, a PROM, an EPROM, an EEPROM, or other similarmemory device.

A phrase, such as “at least one of X, Y, or Z,” unless specificallystated otherwise, is to be understood with the context as used ingeneral to present that an item, term, etc., can be either X, Y, or Z,or any combination thereof (e.g., X, Y, and/or Z). Similarly, “at leastone of X, Y, and Z,” unless specifically stated otherwise, is to beunderstood to present that an item, term, etc., can be either X, Y, andZ, or any combination thereof (e.g., X, Y, and/or Z). Thus, as usedherein, such phases are not generally intended to, and should not, implythat certain embodiments require at least one of either X, Y, or Z to bepresent, but not, for example, one X and one Y. Further, such phasesshould not imply that certain embodiments require each of at least oneof X, at least one of Y, and at least one of Z to be present.

Although embodiments have been described herein in detail, thedescriptions are by way of example. The features of the embodimentsdescribed herein are representative and, in alternative embodiments,certain features and elements may be added or omitted. Additionally,modifications to aspects of the embodiments described herein may be madeby those skilled in the art without departing from the spirit and scopeof the present invention defined in the following claims, the scope ofwhich are to be accorded the broadest interpretation so as to encompassmodifications and equivalent structures.

Therefore, at least the following is claimed:
 1. A local lagged adaptedgeneralized method of moments (LLGMM) process to simulate a forecastusing measured data, the process to simulate comprising: developing astochastic model of a continuous time dynamic process; obtaining adiscrete time data set measured for at least one commodity as past stateinformation of the continuous time dynamic process over a time interval;generating a discrete time interconnected dynamic model of local samplemean and variance statistic processes (DTIDMLSMVSP) based on thestochastic model of the continuous time dynamic process and the discretetime data set measured for at least one commodity; calculating, by atleast one computer, a plurality of admissible parameter estimates forthe stochastic model of the continuous time dynamic process, to forecasta price of the at least one commodity, using the DTIDMLSMVSP; for eachof the plurality of admissible parameter estimates, calculating, by theat least one computer, a state value of the stochastic model of thecontinuous time dynamic process to gather a plurality of state values ofthe stochastic model of the continuous time dynamic process; anddetermining an optimal admissible parameter estimate among the pluralityof admissible parameter estimates that results in a minimum error amongthe plurality of state values, wherein generating the DTIDMLSMVSPfurther comprises: at each time point in a partition of the timeinterval, selecting, by the at least one computer, an m_(k)-pointsub-partition of the partition, the m_(k)-point sub-partition having alocal admissible lagged sample observation size based on an order of amodel, a response delay associated with the continuous time dynamicprocess, and a sub-partition time observation index size; and for eachm_(k)-point in each sub-partition, selecting, by the at least onecomputer, an m_(k)-local moving sequence in the sub-partition to gatheran m_(k)-class of admissible restricted finite sequences.
 2. The LLGMMprocess according to claim 1, wherein generating the DTIDMLSMVSP furthercomprises: for each m_(k)-local moving sequence, calculating, by the atleast one computer, an m_(k)-local average to generate an m_(k)-movingaverage process; and for each m_(k)-local moving sequence, calculating,by the at least one computer, an m_(k)-local variance to generate anm_(k)-local moving variance process.
 3. The LLGMM process according toclaim 2, wherein generating the DTIDMLSMVSP further comprises:transforming the stochastic model of the continuous time dynamic processinto a stochastic model of a discrete time dynamic process utilizing adiscretization scheme; and developing a system of generalized method ofmoments equations from the stochastic model of the discrete time dynamicprocess.
 4. The LLGMM process according to claim 2, further comprisingidentifying an optimal m_(k)-local moving sequence among the m_(k)-classof admissible restricted finite sequences based on the minimum error. 5.The LLGMM process according to claim 4, wherein determining the optimaladmissible parameter estimate comprises: identifying one m_(k)-localmoving sequence among the m-class of admissible restricted finitesequences as the optimal m_(k)-local moving sequence when the onem_(k)-local moving sequence is associated with the minimum error; andselecting a largest m_(k)-local moving sequence among the m_(k)-class ofadmissible restricted finite sequences as the optimal m_(k)-local movingsequence when more than one m_(k)-local moving sequence in them_(k)-class of admissible restricted finite sequences is associated withthe minimum error.
 6. The LLGMM process according to claim 4, furthercomprising forecasting at least one future state value of the stochasticmodel of the continuous-time dynamic process using the optimalm_(k)-local moving sequence.
 7. The LLGMM process according to claim 6,further comprising determining an interval of confidence associated withthe at least one future state value.
 8. A local lagged adaptedgeneralized method of moments (LLGMM) system to simulate a forecastusing measured data, comprising: a memory that stores a discrete timedata set measured for at least one commodity as past state informationof a continuous time dynamic process over a time interval and computerreadable instructions for an LLGMM process; and at least one computingdevice coupled to the memory and configured, through the execution ofthe computer readable instructions for the LLGMM process, to: generate adiscrete time interconnected dynamic model of local sample mean andvariance statistic processes (DTIDMLSMVSP) based on a stochastic modelof a continuous time dynamic process and the discrete time data setmeasured for at least one commodity; calculate a plurality of admissibleparameter estimates for the stochastic model of the continuous timedynamic process, to forecast a price of the at least one commodity,using the DTIDMLSMVSP; for each of the plurality of admissible parameterestimates, calculate a state value of the stochastic model of thecontinuous time dynamic process to gather a plurality of state values ofthe stochastic model of the continuous time dynamic process; determinean optimal admissible parameter estimate among the plurality ofadmissible parameter estimates that results in a minimum error among theplurality of state values; at each time point in a partition of the timeinterval, select an m_(k)-point sub-partition of the partition, them_(k)-point sub-partition having a local admissible lagged sampleobservation size based on an order of a model, a response delayassociated with the continuous time dynamic process, and a sub-partitiontime observation index size; and for each m_(k)-point in eachsub-partition, select an m_(k)-local moving sequence in thesub-partition to gather an m_(k)-class of admissible restricted finitesequences.
 9. The LLGMM system according to claim 8, wherein the atleast one computing device is further configured to: for eachm_(k)-local moving sequence, calculate an mA-local average to generatean m_(k)-moving average process; and for each m_(k)-local movingsequence, calculate an mA-local variance to generate an m_(k)-localmoving variance process.
 10. The LLGMM system according to claim 9,wherein the at least one computing device is further configured to:transform the stochastic model of the continuous time dynamic processinto a stochastic model of a discrete time dynamic process utilizing adiscretization scheme; and develop a system of generalized method ofmoments equations from the stochastic model of a discrete time dynamicprocess.
 11. The LLGMM system according to claim 9, wherein the at leastone computing device is further configured to identify an optimalm_(k)-local moving sequence among the m_(k)-class of admissiblerestricted finite sequences based on the minimum error.
 12. The LLGMMsystem according to claim 11, wherein the at least one computing deviceis further configured to: identify one m_(k)-local moving sequence amongthe m_(k)-class of admissible restricted finite sequences as the optimalm_(k)-local moving sequence when the one m_(k)-local moving sequence isassociated with the minimum error; and select a largest m_(k)-localmoving sequence among the m-class of admissible restricted finitesequences as the optimal m_(k)-local moving sequence when more than onem_(k)-local moving sequence in the m_(k)-class of admissible restrictedfinite sequences is associated with the minimum error.
 13. The LLGMMprocess according to claim 11, wherein the at least one computing deviceis further configured to forecast at least one future state value of thestochastic model of the continuous-time dynamic process using theoptimal m_(k)-local moving sequence.
 14. A non-transitory computerreadable medium including computer readable instructions stored thereonthat, when executed by at least one computing device, direct the atleast one computing device to perform a local lagged adapted generalizedmethod of moments (LLGMM) process to simulate a forecast using measureddata, the process to simulate comprising: obtaining a discrete time dataset measured for at least one commodity as past state information of acontinuous time dynamic process over a time interval; generating adiscrete time interconnected dynamic model of local sample mean andvariance statistic processes (DTIDMLSMVSP) based on a stochastic modelof a continuous time dynamic process and the discrete time data setmeasured for at least one commodity; calculating, by the at least onecomputing device, a plurality of admissible parameter estimates for thestochastic model of the continuous time dynamic process, to forecast aprice of the at least one commodity, using the DTIDMLSMVSP; for each ofthe plurality of admissible parameter estimates, calculating, by the atleast one computer, a state value of the stochastic model of thecontinuous time dynamic process to gather a plurality of state values ofthe stochastic model of the continuous time dynamic process; anddetermining an optimal admissible parameter estimate among the pluralityof admissible parameter estimates that results in a minimum error amongthe plurality of state values, wherein generating the DTIDMLSMVSPfurther comprises: at each time point in a partition of the timeinterval, selecting, by at least one computer, an m_(k)-pointsub-partition of the partition, the m_(k)-point sub-partition having alocal admissible lagged sample observation size based on an order of amodel, a response delay associated with the continuous time dynamicprocess, and a sub-partition time observation index size; for eachm_(k)-point in each sub-partition, selecting, by the at least onecomputer, an m_(k)-local moving sequence in the sub-partition to gatheran m_(k)-class of admissible restricted finite sequences; for eachm_(k)-local moving sequence, calculating, by the at least one computer,an m_(k)-local average to generate an m_(k)-moving average process; andfor each m_(k)-local moving sequence, calculating, by the at least onecomputer, an m_(k)-local variance to generate an m_(k)-local movingvariance process.